Examples
A=[350750110]A=⎡⎢⎣350750110⎤⎥⎦
Step 1
Step 1.1
Set up the formula to find the characteristic equation p(λ)p(λ).
p(λ)=determinant(A-λI3)p(λ)=determinant(A−λI3)
Step 1.2
The identity matrix or unit matrix of size 33 is the 3×33×3 square matrix with ones on the main diagonal and zeros elsewhere.
[100010001]⎡⎢⎣100010001⎤⎥⎦
Step 1.3
Substitute the known values into p(λ)=determinant(A-λI3)p(λ)=determinant(A−λI3).
Step 1.3.1
Substitute [350750110]⎡⎢⎣350750110⎤⎥⎦ for AA.
p(λ)=determinant([350750110]-λI3)p(λ)=determinant⎛⎜⎝⎡⎢⎣350750110⎤⎥⎦−λI3⎞⎟⎠
Step 1.3.2
Substitute [100010001]⎡⎢⎣100010001⎤⎥⎦ for I3I3.
p(λ)=determinant([350750110]-λ[100010001])p(λ)=determinant⎛⎜⎝⎡⎢⎣350750110⎤⎥⎦−λ⎡⎢⎣100010001⎤⎥⎦⎞⎟⎠
p(λ)=determinant([350750110]-λ[100010001])p(λ)=determinant⎛⎜⎝⎡⎢⎣350750110⎤⎥⎦−λ⎡⎢⎣100010001⎤⎥⎦⎞⎟⎠
Step 1.4
Simplify.
Step 1.4.1
Simplify each term.
Step 1.4.1.1
Multiply -λ−λ by each element of the matrix.
p(λ)=determinant([350750110]+[-λ⋅1-λ⋅0-λ⋅0-λ⋅0-λ⋅1-λ⋅0-λ⋅0-λ⋅0-λ⋅1])p(λ)=determinant⎛⎜⎝⎡⎢⎣350750110⎤⎥⎦+⎡⎢⎣−λ⋅1−λ⋅0−λ⋅0−λ⋅0−λ⋅1−λ⋅0−λ⋅0−λ⋅0−λ⋅1⎤⎥⎦⎞⎟⎠
Step 1.4.1.2
Simplify each element in the matrix.
Step 1.4.1.2.1
Multiply -1−1 by 11.
p(λ)=determinant([350750110]+[-λ-λ⋅0-λ⋅0-λ⋅0-λ⋅1-λ⋅0-λ⋅0-λ⋅0-λ⋅1])p(λ)=determinant⎛⎜⎝⎡⎢⎣350750110⎤⎥⎦+⎡⎢⎣−λ−λ⋅0−λ⋅0−λ⋅0−λ⋅1−λ⋅0−λ⋅0−λ⋅0−λ⋅1⎤⎥⎦⎞⎟⎠
Step 1.4.1.2.2
Multiply -λ⋅0−λ⋅0.
Step 1.4.1.2.2.1
Multiply 00 by -1−1.
p(λ)=determinant([350750110]+[-λ0λ-λ⋅0-λ⋅0-λ⋅1-λ⋅0-λ⋅0-λ⋅0-λ⋅1])p(λ)=determinant⎛⎜⎝⎡⎢⎣350750110⎤⎥⎦+⎡⎢⎣−λ0λ−λ⋅0−λ⋅0−λ⋅1−λ⋅0−λ⋅0−λ⋅0−λ⋅1⎤⎥⎦⎞⎟⎠
Step 1.4.1.2.2.2
Multiply 00 by λλ.
p(λ)=determinant([350750110]+[-λ0-λ⋅0-λ⋅0-λ⋅1-λ⋅0-λ⋅0-λ⋅0-λ⋅1])p(λ)=determinant⎛⎜⎝⎡⎢⎣350750110⎤⎥⎦+⎡⎢⎣−λ0−λ⋅0−λ⋅0−λ⋅1−λ⋅0−λ⋅0−λ⋅0−λ⋅1⎤⎥⎦⎞⎟⎠
p(λ)=determinant([350750110]+[-λ0-λ⋅0-λ⋅0-λ⋅1-λ⋅0-λ⋅0-λ⋅0-λ⋅1])p(λ)=determinant⎛⎜⎝⎡⎢⎣350750110⎤⎥⎦+⎡⎢⎣−λ0−λ⋅0−λ⋅0−λ⋅1−λ⋅0−λ⋅0−λ⋅0−λ⋅1⎤⎥⎦⎞⎟⎠
Step 1.4.1.2.3
Multiply -λ⋅0−λ⋅0.
Step 1.4.1.2.3.1
Multiply 00 by -1−1.
p(λ)=determinant([350750110]+[-λ00λ-λ⋅0-λ⋅1-λ⋅0-λ⋅0-λ⋅0-λ⋅1])p(λ)=determinant⎛⎜⎝⎡⎢⎣350750110⎤⎥⎦+⎡⎢⎣−λ00λ−λ⋅0−λ⋅1−λ⋅0−λ⋅0−λ⋅0−λ⋅1⎤⎥⎦⎞⎟⎠
Step 1.4.1.2.3.2
Multiply 00 by λλ.
p(λ)=determinant([350750110]+[-λ00-λ⋅0-λ⋅1-λ⋅0-λ⋅0-λ⋅0-λ⋅1])p(λ)=determinant⎛⎜⎝⎡⎢⎣350750110⎤⎥⎦+⎡⎢⎣−λ00−λ⋅0−λ⋅1−λ⋅0−λ⋅0−λ⋅0−λ⋅1⎤⎥⎦⎞⎟⎠
p(λ)=determinant([350750110]+[-λ00-λ⋅0-λ⋅1-λ⋅0-λ⋅0-λ⋅0-λ⋅1])p(λ)=determinant⎛⎜⎝⎡⎢⎣350750110⎤⎥⎦+⎡⎢⎣−λ00−λ⋅0−λ⋅1−λ⋅0−λ⋅0−λ⋅0−λ⋅1⎤⎥⎦⎞⎟⎠
Step 1.4.1.2.4
Multiply -λ⋅0−λ⋅0.
Step 1.4.1.2.4.1
Multiply 00 by -1−1.
p(λ)=determinant([350750110]+[-λ000λ-λ⋅1-λ⋅0-λ⋅0-λ⋅0-λ⋅1])p(λ)=determinant⎛⎜⎝⎡⎢⎣350750110⎤⎥⎦+⎡⎢⎣−λ000λ−λ⋅1−λ⋅0−λ⋅0−λ⋅0−λ⋅1⎤⎥⎦⎞⎟⎠
Step 1.4.1.2.4.2
Multiply 00 by λλ.
p(λ)=determinant([350750110]+[-λ000-λ⋅1-λ⋅0-λ⋅0-λ⋅0-λ⋅1])p(λ)=determinant⎛⎜⎝⎡⎢⎣350750110⎤⎥⎦+⎡⎢⎣−λ000−λ⋅1−λ⋅0−λ⋅0−λ⋅0−λ⋅1⎤⎥⎦⎞⎟⎠
p(λ)=determinant([350750110]+[-λ000-λ⋅1-λ⋅0-λ⋅0-λ⋅0-λ⋅1])p(λ)=determinant⎛⎜⎝⎡⎢⎣350750110⎤⎥⎦+⎡⎢⎣−λ000−λ⋅1−λ⋅0−λ⋅0−λ⋅0−λ⋅1⎤⎥⎦⎞⎟⎠
Step 1.4.1.2.5
Multiply -1−1 by 11.
p(λ)=determinant([350750110]+[-λ000-λ-λ⋅0-λ⋅0-λ⋅0-λ⋅1])p(λ)=determinant⎛⎜⎝⎡⎢⎣350750110⎤⎥⎦+⎡⎢⎣−λ000−λ−λ⋅0−λ⋅0−λ⋅0−λ⋅1⎤⎥⎦⎞⎟⎠
Step 1.4.1.2.6
Multiply -λ⋅0−λ⋅0.
Step 1.4.1.2.6.1
Multiply 00 by -1−1.
p(λ)=determinant([350750110]+[-λ000-λ0λ-λ⋅0-λ⋅0-λ⋅1])p(λ)=determinant⎛⎜⎝⎡⎢⎣350750110⎤⎥⎦+⎡⎢⎣−λ000−λ0λ−λ⋅0−λ⋅0−λ⋅1⎤⎥⎦⎞⎟⎠
Step 1.4.1.2.6.2
Multiply 00 by λλ.
p(λ)=determinant([350750110]+[-λ000-λ0-λ⋅0-λ⋅0-λ⋅1])p(λ)=determinant⎛⎜⎝⎡⎢⎣350750110⎤⎥⎦+⎡⎢⎣−λ000−λ0−λ⋅0−λ⋅0−λ⋅1⎤⎥⎦⎞⎟⎠
p(λ)=determinant([350750110]+[-λ000-λ0-λ⋅0-λ⋅0-λ⋅1])p(λ)=determinant⎛⎜⎝⎡⎢⎣350750110⎤⎥⎦+⎡⎢⎣−λ000−λ0−λ⋅0−λ⋅0−λ⋅1⎤⎥⎦⎞⎟⎠
Step 1.4.1.2.7
Multiply -λ⋅0−λ⋅0.
Step 1.4.1.2.7.1
Multiply 00 by -1−1.
p(λ)=determinant([350750110]+[-λ000-λ00λ-λ⋅0-λ⋅1])p(λ)=determinant⎛⎜⎝⎡⎢⎣350750110⎤⎥⎦+⎡⎢⎣−λ000−λ00λ−λ⋅0−λ⋅1⎤⎥⎦⎞⎟⎠
Step 1.4.1.2.7.2
Multiply 00 by λλ.
p(λ)=determinant([350750110]+[-λ000-λ00-λ⋅0-λ⋅1])p(λ)=determinant⎛⎜⎝⎡⎢⎣350750110⎤⎥⎦+⎡⎢⎣−λ000−λ00−λ⋅0−λ⋅1⎤⎥⎦⎞⎟⎠
p(λ)=determinant([350750110]+[-λ000-λ00-λ⋅0-λ⋅1])p(λ)=determinant⎛⎜⎝⎡⎢⎣350750110⎤⎥⎦+⎡⎢⎣−λ000−λ00−λ⋅0−λ⋅1⎤⎥⎦⎞⎟⎠
Step 1.4.1.2.8
Multiply -λ⋅0−λ⋅0.
Step 1.4.1.2.8.1
Multiply 00 by -1−1.
p(λ)=determinant([350750110]+[-λ000-λ000λ-λ⋅1])p(λ)=determinant⎛⎜⎝⎡⎢⎣350750110⎤⎥⎦+⎡⎢⎣−λ000−λ000λ−λ⋅1⎤⎥⎦⎞⎟⎠
Step 1.4.1.2.8.2
Multiply 00 by λλ.
p(λ)=determinant([350750110]+[-λ000-λ000-λ⋅1])p(λ)=determinant⎛⎜⎝⎡⎢⎣350750110⎤⎥⎦+⎡⎢⎣−λ000−λ000−λ⋅1⎤⎥⎦⎞⎟⎠
p(λ)=determinant([350750110]+[-λ000-λ000-λ⋅1])p(λ)=determinant⎛⎜⎝⎡⎢⎣350750110⎤⎥⎦+⎡⎢⎣−λ000−λ000−λ⋅1⎤⎥⎦⎞⎟⎠
Step 1.4.1.2.9
Multiply -1−1 by 11.
p(λ)=determinant([350750110]+[-λ000-λ000-λ])p(λ)=determinant⎛⎜⎝⎡⎢⎣350750110⎤⎥⎦+⎡⎢⎣−λ000−λ000−λ⎤⎥⎦⎞⎟⎠
p(λ)=determinant([350750110]+[-λ000-λ000-λ])p(λ)=determinant⎛⎜⎝⎡⎢⎣350750110⎤⎥⎦+⎡⎢⎣−λ000−λ000−λ⎤⎥⎦⎞⎟⎠
p(λ)=determinant([350750110]+[-λ000-λ000-λ])p(λ)=determinant⎛⎜⎝⎡⎢⎣350750110⎤⎥⎦+⎡⎢⎣−λ000−λ000−λ⎤⎥⎦⎞⎟⎠
Step 1.4.2
Add the corresponding elements.
p(λ)=determinant[3-λ5+00+07+05-λ0+01+01+00-λ]p(λ)=determinant⎡⎢⎣3−λ5+00+07+05−λ0+01+01+00−λ⎤⎥⎦
Step 1.4.3
Simplify each element.
Step 1.4.3.1
Add 55 and 00.
p(λ)=determinant[3-λ50+07+05-λ0+01+01+00-λ]p(λ)=determinant⎡⎢⎣3−λ50+07+05−λ0+01+01+00−λ⎤⎥⎦
Step 1.4.3.2
Add 00 and 00.
p(λ)=determinant[3-λ507+05-λ0+01+01+00-λ]p(λ)=determinant⎡⎢⎣3−λ507+05−λ0+01+01+00−λ⎤⎥⎦
Step 1.4.3.3
Add 77 and 00.
p(λ)=determinant[3-λ5075-λ0+01+01+00-λ]p(λ)=determinant⎡⎢⎣3−λ5075−λ0+01+01+00−λ⎤⎥⎦
Step 1.4.3.4
Add 00 and 00.
p(λ)=determinant[3-λ5075-λ01+01+00-λ]p(λ)=determinant⎡⎢⎣3−λ5075−λ01+01+00−λ⎤⎥⎦
Step 1.4.3.5
Add 11 and 00.
p(λ)=determinant[3-λ5075-λ011+00-λ]p(λ)=determinant⎡⎢⎣3−λ5075−λ011+00−λ⎤⎥⎦
Step 1.4.3.6
Add 11 and 00.
p(λ)=determinant[3-λ5075-λ0110-λ]p(λ)=determinant⎡⎢⎣3−λ5075−λ0110−λ⎤⎥⎦
Step 1.4.3.7
Subtract λλ from 00.
p(λ)=determinant[3-λ5075-λ011-λ]p(λ)=determinant⎡⎢⎣3−λ5075−λ011−λ⎤⎥⎦
p(λ)=determinant[3-λ5075-λ011-λ]p(λ)=determinant⎡⎢⎣3−λ5075−λ011−λ⎤⎥⎦
p(λ)=determinant[3-λ5075-λ011-λ]p(λ)=determinant⎡⎢⎣3−λ5075−λ011−λ⎤⎥⎦
Step 1.5
Find the determinant.
Step 1.5.1
Choose the row or column with the most 00 elements. If there are no 00 elements choose any row or column. Multiply every element in column 33 by its cofactor and add.
Step 1.5.1.1
Consider the corresponding sign chart.
|+-+-+-+-+|∣∣
∣∣+−+−+−+−+∣∣
∣∣
Step 1.5.1.2
The cofactor is the minor with the sign changed if the indices match a -− position on the sign chart.
Step 1.5.1.3
The minor for a13a13 is the determinant with row 11 and column 33 deleted.
|75-λ11|∣∣∣75−λ11∣∣∣
Step 1.5.1.4
Multiply element a13a13 by its cofactor.
0|75-λ11|0∣∣∣75−λ11∣∣∣
Step 1.5.1.5
The minor for a23a23 is the determinant with row 22 and column 33 deleted.
|3-λ511|∣∣∣3−λ511∣∣∣
Step 1.5.1.6
Multiply element a23a23 by its cofactor.
0|3-λ511|0∣∣∣3−λ511∣∣∣
Step 1.5.1.7
The minor for a33a33 is the determinant with row 33 and column 33 deleted.
|3-λ575-λ|∣∣∣3−λ575−λ∣∣∣
Step 1.5.1.8
Multiply element a33a33 by its cofactor.
-λ|3-λ575-λ|−λ∣∣∣3−λ575−λ∣∣∣
Step 1.5.1.9
Add the terms together.
p(λ)=0|75-λ11|+0|3-λ511|-λ|3-λ575-λ|p(λ)=0∣∣∣75−λ11∣∣∣+0∣∣∣3−λ511∣∣∣−λ∣∣∣3−λ575−λ∣∣∣
p(λ)=0|75-λ11|+0|3-λ511|-λ|3-λ575-λ|p(λ)=0∣∣∣75−λ11∣∣∣+0∣∣∣3−λ511∣∣∣−λ∣∣∣3−λ575−λ∣∣∣
Step 1.5.2
Multiply 00 by |75-λ11|∣∣∣75−λ11∣∣∣.
p(λ)=0+0|3-λ511|-λ|3-λ575-λ|p(λ)=0+0∣∣∣3−λ511∣∣∣−λ∣∣∣3−λ575−λ∣∣∣
Step 1.5.3
Multiply 00 by |3-λ511|∣∣∣3−λ511∣∣∣.
p(λ)=0+0-λ|3-λ575-λ|p(λ)=0+0−λ∣∣∣3−λ575−λ∣∣∣
Step 1.5.4
Evaluate |3-λ575-λ|∣∣∣3−λ575−λ∣∣∣.
Step 1.5.4.1
The determinant of a 2×22×2 matrix can be found using the formula |abcd|=ad-cb∣∣∣abcd∣∣∣=ad−cb.
p(λ)=0+0-λ((3-λ)(5-λ)-7⋅5)p(λ)=0+0−λ((3−λ)(5−λ)−7⋅5)
Step 1.5.4.2
Simplify the determinant.
Step 1.5.4.2.1
Simplify each term.
Step 1.5.4.2.1.1
Expand (3-λ)(5-λ)(3−λ)(5−λ) using the FOIL Method.
Step 1.5.4.2.1.1.1
Apply the distributive property.
p(λ)=0+0-λ(3(5-λ)-λ(5-λ)-7⋅5)p(λ)=0+0−λ(3(5−λ)−λ(5−λ)−7⋅5)
Step 1.5.4.2.1.1.2
Apply the distributive property.
p(λ)=0+0-λ(3⋅5+3(-λ)-λ(5-λ)-7⋅5)p(λ)=0+0−λ(3⋅5+3(−λ)−λ(5−λ)−7⋅5)
Step 1.5.4.2.1.1.3
Apply the distributive property.
p(λ)=0+0-λ(3⋅5+3(-λ)-λ⋅5-λ(-λ)-7⋅5)p(λ)=0+0−λ(3⋅5+3(−λ)−λ⋅5−λ(−λ)−7⋅5)
p(λ)=0+0-λ(3⋅5+3(-λ)-λ⋅5-λ(-λ)-7⋅5)p(λ)=0+0−λ(3⋅5+3(−λ)−λ⋅5−λ(−λ)−7⋅5)
Step 1.5.4.2.1.2
Simplify and combine like terms.
Step 1.5.4.2.1.2.1
Simplify each term.
Step 1.5.4.2.1.2.1.1
Multiply 33 by 55.
p(λ)=0+0-λ(15+3(-λ)-λ⋅5-λ(-λ)-7⋅5)p(λ)=0+0−λ(15+3(−λ)−λ⋅5−λ(−λ)−7⋅5)
Step 1.5.4.2.1.2.1.2
Multiply -1−1 by 33.
p(λ)=0+0-λ(15-3λ-λ⋅5-λ(-λ)-7⋅5)p(λ)=0+0−λ(15−3λ−λ⋅5−λ(−λ)−7⋅5)
Step 1.5.4.2.1.2.1.3
Multiply 55 by -1−1.
p(λ)=0+0-λ(15-3λ-5λ-λ(-λ)-7⋅5)p(λ)=0+0−λ(15−3λ−5λ−λ(−λ)−7⋅5)
Step 1.5.4.2.1.2.1.4
Rewrite using the commutative property of multiplication.
p(λ)=0+0-λ(15-3λ-5λ-1⋅-1λ⋅λ-7⋅5)p(λ)=0+0−λ(15−3λ−5λ−1⋅−1λ⋅λ−7⋅5)
Step 1.5.4.2.1.2.1.5
Multiply λλ by λλ by adding the exponents.
Step 1.5.4.2.1.2.1.5.1
Move λλ.
p(λ)=0+0-λ(15-3λ-5λ-1⋅-1(λ⋅λ)-7⋅5)p(λ)=0+0−λ(15−3λ−5λ−1⋅−1(λ⋅λ)−7⋅5)
Step 1.5.4.2.1.2.1.5.2
Multiply λλ by λλ.
p(λ)=0+0-λ(15-3λ-5λ-1⋅-1λ2-7⋅5)p(λ)=0+0−λ(15−3λ−5λ−1⋅−1λ2−7⋅5)
p(λ)=0+0-λ(15-3λ-5λ-1⋅-1λ2-7⋅5)p(λ)=0+0−λ(15−3λ−5λ−1⋅−1λ2−7⋅5)
Step 1.5.4.2.1.2.1.6
Multiply -1−1 by -1−1.
p(λ)=0+0-λ(15-3λ-5λ+1λ2-7⋅5)p(λ)=0+0−λ(15−3λ−5λ+1λ2−7⋅5)
Step 1.5.4.2.1.2.1.7
Multiply λ2λ2 by 11.
p(λ)=0+0-λ(15-3λ-5λ+λ2-7⋅5)p(λ)=0+0−λ(15−3λ−5λ+λ2−7⋅5)
p(λ)=0+0-λ(15-3λ-5λ+λ2-7⋅5)p(λ)=0+0−λ(15−3λ−5λ+λ2−7⋅5)
Step 1.5.4.2.1.2.2
Subtract 5λ5λ from -3λ−3λ.
p(λ)=0+0-λ(15-8λ+λ2-7⋅5)p(λ)=0+0−λ(15−8λ+λ2−7⋅5)
p(λ)=0+0-λ(15-8λ+λ2-7⋅5)p(λ)=0+0−λ(15−8λ+λ2−7⋅5)
Step 1.5.4.2.1.3
Multiply -7−7 by 55.
p(λ)=0+0-λ(15-8λ+λ2-35)p(λ)=0+0−λ(15−8λ+λ2−35)
p(λ)=0+0-λ(15-8λ+λ2-35)p(λ)=0+0−λ(15−8λ+λ2−35)
Step 1.5.4.2.2
Subtract 3535 from 1515.
p(λ)=0+0-λ(-8λ+λ2-20)p(λ)=0+0−λ(−8λ+λ2−20)
Step 1.5.4.2.3
Reorder -8λ−8λ and λ2λ2.
p(λ)=0+0-λ(λ2-8λ-20)p(λ)=0+0−λ(λ2−8λ−20)
p(λ)=0+0-λ(λ2-8λ-20)p(λ)=0+0−λ(λ2−8λ−20)
p(λ)=0+0-λ(λ2-8λ-20)p(λ)=0+0−λ(λ2−8λ−20)
Step 1.5.5
Simplify the determinant.
Step 1.5.5.1
Combine the opposite terms in 0+0-λ(λ2-8λ-20)0+0−λ(λ2−8λ−20).
Step 1.5.5.1.1
Add 00 and 00.
p(λ)=0-λ(λ2-8λ-20)p(λ)=0−λ(λ2−8λ−20)
Step 1.5.5.1.2
Subtract λ(λ2-8λ-20)λ(λ2−8λ−20) from 00.
p(λ)=-λ(λ2-8λ-20)p(λ)=−λ(λ2−8λ−20)
p(λ)=-λ(λ2-8λ-20)p(λ)=−λ(λ2−8λ−20)
Step 1.5.5.2
Apply the distributive property.
p(λ)=-λ⋅λ2-λ(-8λ)-λ⋅-20p(λ)=−λ⋅λ2−λ(−8λ)−λ⋅−20
Step 1.5.5.3
Simplify.
Step 1.5.5.3.1
Multiply λλ by λ2λ2 by adding the exponents.
Step 1.5.5.3.1.1
Move λ2λ2.
p(λ)=-(λ2λ)-λ(-8λ)-λ⋅-20p(λ)=−(λ2λ)−λ(−8λ)−λ⋅−20
Step 1.5.5.3.1.2
Multiply λ2λ2 by λλ.
Step 1.5.5.3.1.2.1
Raise λλ to the power of 11.
p(λ)=-(λ2λ1)-λ(-8λ)-λ⋅-20p(λ)=−(λ2λ1)−λ(−8λ)−λ⋅−20
Step 1.5.5.3.1.2.2
Use the power rule aman=am+naman=am+n to combine exponents.
p(λ)=-λ2+1-λ(-8λ)-λ⋅-20p(λ)=−λ2+1−λ(−8λ)−λ⋅−20
p(λ)=-λ2+1-λ(-8λ)-λ⋅-20p(λ)=−λ2+1−λ(−8λ)−λ⋅−20
Step 1.5.5.3.1.3
Add 22 and 11.
p(λ)=-λ3-λ(-8λ)-λ⋅-20p(λ)=−λ3−λ(−8λ)−λ⋅−20
p(λ)=-λ3-λ(-8λ)-λ⋅-20p(λ)=−λ3−λ(−8λ)−λ⋅−20
Step 1.5.5.3.2
Rewrite using the commutative property of multiplication.
p(λ)=-λ3-1⋅-8λ⋅λ-λ⋅-20p(λ)=−λ3−1⋅−8λ⋅λ−λ⋅−20
Step 1.5.5.3.3
Multiply -20−20 by -1−1.
p(λ)=-λ3-1⋅-8λ⋅λ+20λp(λ)=−λ3−1⋅−8λ⋅λ+20λ
p(λ)=-λ3-1⋅-8λ⋅λ+20λp(λ)=−λ3−1⋅−8λ⋅λ+20λ
Step 1.5.5.4
Simplify each term.
Step 1.5.5.4.1
Multiply λλ by λλ by adding the exponents.
Step 1.5.5.4.1.1
Move λλ.
p(λ)=-λ3-1⋅-8(λ⋅λ)+20λp(λ)=−λ3−1⋅−8(λ⋅λ)+20λ
Step 1.5.5.4.1.2
Multiply λλ by λλ.
p(λ)=-λ3-1⋅-8λ2+20λp(λ)=−λ3−1⋅−8λ2+20λ
p(λ)=-λ3-1⋅-8λ2+20λp(λ)=−λ3−1⋅−8λ2+20λ
Step 1.5.5.4.2
Multiply -1−1 by -8−8.
p(λ)=-λ3+8λ2+20λp(λ)=−λ3+8λ2+20λ
p(λ)=-λ3+8λ2+20λp(λ)=−λ3+8λ2+20λ
p(λ)=-λ3+8λ2+20λp(λ)=−λ3+8λ2+20λ
p(λ)=-λ3+8λ2+20λp(λ)=−λ3+8λ2+20λ
Step 1.6
Set the characteristic polynomial equal to 00 to find the eigenvalues λλ.
-λ3+8λ2+20λ=0−λ3+8λ2+20λ=0
Step 1.7
Solve for λλ.
Step 1.7.1
Factor the left side of the equation.
Step 1.7.1.1
Factor -λ−λ out of -λ3+8λ2+20λ−λ3+8λ2+20λ.
Step 1.7.1.1.1
Factor -λ−λ out of -λ3−λ3.
-λ⋅λ2+8λ2+20λ=0−λ⋅λ2+8λ2+20λ=0
Step 1.7.1.1.2
Factor -λ−λ out of 8λ28λ2.
-λ⋅λ2-λ(-8λ)+20λ=0−λ⋅λ2−λ(−8λ)+20λ=0
Step 1.7.1.1.3
Factor -λ−λ out of 20λ20λ.
-λ⋅λ2-λ(-8λ)-λ⋅-20=0−λ⋅λ2−λ(−8λ)−λ⋅−20=0
Step 1.7.1.1.4
Factor -λ−λ out of -λ(λ2)-λ(-8λ)−λ(λ2)−λ(−8λ).
-λ(λ2-8λ)-λ⋅-20=0−λ(λ2−8λ)−λ⋅−20=0
Step 1.7.1.1.5
Factor -λ−λ out of -λ(λ2-8λ)-λ(-20)−λ(λ2−8λ)−λ(−20).
-λ(λ2-8λ-20)=0−λ(λ2−8λ−20)=0
-λ(λ2-8λ-20)=0−λ(λ2−8λ−20)=0
Step 1.7.1.2
Factor.
Step 1.7.1.2.1
Factor λ2-8λ-20λ2−8λ−20 using the AC method.
Step 1.7.1.2.1.1
Consider the form x2+bx+cx2+bx+c. Find a pair of integers whose product is cc and whose sum is bb. In this case, whose product is -20−20 and whose sum is -8−8.
-10,2−10,2
Step 1.7.1.2.1.2
Write the factored form using these integers.
-λ((λ-10)(λ+2))=0−λ((λ−10)(λ+2))=0
-λ((λ-10)(λ+2))=0−λ((λ−10)(λ+2))=0
Step 1.7.1.2.2
Remove unnecessary parentheses.
-λ(λ-10)(λ+2)=0−λ(λ−10)(λ+2)=0
-λ(λ-10)(λ+2)=0−λ(λ−10)(λ+2)=0
-λ(λ-10)(λ+2)=0−λ(λ−10)(λ+2)=0
Step 1.7.2
If any individual factor on the left side of the equation is equal to 00, the entire expression will be equal to 00.
λ=0λ=0
λ-10=0λ−10=0
λ+2=0λ+2=0
Step 1.7.3
Set λλ equal to 00.
λ=0λ=0
Step 1.7.4
Set λ-10λ−10 equal to 00 and solve for λλ.
Step 1.7.4.1
Set λ-10λ−10 equal to 00.
λ-10=0λ−10=0
Step 1.7.4.2
Add 1010 to both sides of the equation.
λ=10λ=10
λ=10λ=10
Step 1.7.5
Set λ+2λ+2 equal to 00 and solve for λλ.
Step 1.7.5.1
Set λ+2λ+2 equal to 00.
λ+2=0λ+2=0
Step 1.7.5.2
Subtract 22 from both sides of the equation.
λ=-2λ=−2
λ=-2λ=−2
Step 1.7.6
The final solution is all the values that make -λ(λ-10)(λ+2)=0−λ(λ−10)(λ+2)=0 true.
λ=0,10,-2λ=0,10,−2
λ=0,10,-2λ=0,10,−2
λ=0,10,-2λ=0,10,−2
Step 2
The eigenvector is equal to the null space of the matrix minus the eigenvalue times the identity matrix where NN is the null space and II is the identity matrix.
εA=N(A-λI3)εA=N(A−λI3)
Step 3
Step 3.1
Substitute the known values into the formula.
N([350750110]+0[100010001])N⎛⎜⎝⎡⎢⎣350750110⎤⎥⎦+0⎡⎢⎣100010001⎤⎥⎦⎞⎟⎠
Step 3.2
Simplify.
Step 3.2.1
Simplify each term.
Step 3.2.1.1
Multiply 00 by each element of the matrix.
[350750110]+[0⋅10⋅00⋅00⋅00⋅10⋅00⋅00⋅00⋅1]⎡⎢⎣350750110⎤⎥⎦+⎡⎢⎣0⋅10⋅00⋅00⋅00⋅10⋅00⋅00⋅00⋅1⎤⎥⎦
Step 3.2.1.2
Simplify each element in the matrix.
Step 3.2.1.2.1
Multiply 00 by 11.
[350750110]+[00⋅00⋅00⋅00⋅10⋅00⋅00⋅00⋅1]⎡⎢⎣350750110⎤⎥⎦+⎡⎢⎣00⋅00⋅00⋅00⋅10⋅00⋅00⋅00⋅1⎤⎥⎦
Step 3.2.1.2.2
Multiply 00 by 00.
[350750110]+[000⋅00⋅00⋅10⋅00⋅00⋅00⋅1]⎡⎢⎣350750110⎤⎥⎦+⎡⎢⎣000⋅00⋅00⋅10⋅00⋅00⋅00⋅1⎤⎥⎦
Step 3.2.1.2.3
Multiply 00 by 00.
[350750110]+[0000⋅00⋅10⋅00⋅00⋅00⋅1]⎡⎢⎣350750110⎤⎥⎦+⎡⎢⎣0000⋅00⋅10⋅00⋅00⋅00⋅1⎤⎥⎦
Step 3.2.1.2.4
Multiply 00 by 00.
[350750110]+[00000⋅10⋅00⋅00⋅00⋅1]⎡⎢⎣350750110⎤⎥⎦+⎡⎢⎣00000⋅10⋅00⋅00⋅00⋅1⎤⎥⎦
Step 3.2.1.2.5
Multiply 00 by 11.
[350750110]+[000000⋅00⋅00⋅00⋅1]⎡⎢⎣350750110⎤⎥⎦+⎡⎢⎣000000⋅00⋅00⋅00⋅1⎤⎥⎦
Step 3.2.1.2.6
Multiply 00 by 00.
[350750110]+[0000000⋅00⋅00⋅1]⎡⎢⎣350750110⎤⎥⎦+⎡⎢⎣0000000⋅00⋅00⋅1⎤⎥⎦
Step 3.2.1.2.7
Multiply 00 by 00.
[350750110]+[00000000⋅00⋅1]⎡⎢⎣350750110⎤⎥⎦+⎡⎢⎣00000000⋅00⋅1⎤⎥⎦
Step 3.2.1.2.8
Multiply 00 by 00.
[350750110]+[000000000⋅1]⎡⎢⎣350750110⎤⎥⎦+⎡⎢⎣000000000⋅1⎤⎥⎦
Step 3.2.1.2.9
Multiply 00 by 11.
[350750110]+[000000000]⎡⎢⎣350750110⎤⎥⎦+⎡⎢⎣000000000⎤⎥⎦
[350750110]+[000000000]⎡⎢⎣350750110⎤⎥⎦+⎡⎢⎣000000000⎤⎥⎦
[350750110]+[000000000]⎡⎢⎣350750110⎤⎥⎦+⎡⎢⎣000000000⎤⎥⎦
Step 3.2.2
Adding any matrix to a null matrix is the matrix itself.
Step 3.2.2.1
Add the corresponding elements.
[3+05+00+07+05+00+01+01+00+0]⎡⎢⎣3+05+00+07+05+00+01+01+00+0⎤⎥⎦
Step 3.2.2.2
Simplify each element.
Step 3.2.2.2.1
Add 33 and 00.
[35+00+07+05+00+01+01+00+0]⎡⎢⎣35+00+07+05+00+01+01+00+0⎤⎥⎦
Step 3.2.2.2.2
Add 55 and 00.
[350+07+05+00+01+01+00+0]⎡⎢⎣350+07+05+00+01+01+00+0⎤⎥⎦
Step 3.2.2.2.3
Add 00 and 00.
[3507+05+00+01+01+00+0]⎡⎢⎣3507+05+00+01+01+00+0⎤⎥⎦
Step 3.2.2.2.4
Add 77 and 00.
[35075+00+01+01+00+0]⎡⎢⎣35075+00+01+01+00+0⎤⎥⎦
Step 3.2.2.2.5
Add 55 and 00.
[350750+01+01+00+0]⎡⎢⎣350750+01+01+00+0⎤⎥⎦
Step 3.2.2.2.6
Add 00 and 00.
[3507501+01+00+0]⎡⎢⎣3507501+01+00+0⎤⎥⎦
Step 3.2.2.2.7
Add 11 and 00.
[35075011+00+0]⎡⎢⎣35075011+00+0⎤⎥⎦
Step 3.2.2.2.8
Add 11 and 00.
[350750110+0]⎡⎢⎣350750110+0⎤⎥⎦
Step 3.2.2.2.9
Add 00 and 00.
[350750110]⎡⎢⎣350750110⎤⎥⎦
[350750110]⎡⎢⎣350750110⎤⎥⎦
[350750110]⎡⎢⎣350750110⎤⎥⎦
[350750110]⎡⎢⎣350750110⎤⎥⎦
Step 3.3
Find the null space when λ=0λ=0.
Step 3.3.1
Write as an augmented matrix for Ax=0Ax=0.
[350075001100]⎡⎢
⎢⎣350075001100⎤⎥
⎥⎦
Step 3.3.2
Find the reduced row echelon form.
Step 3.3.2.1
Multiply each element of R1R1 by 1313 to make the entry at 1,11,1 a 11.
Step 3.3.2.1.1
Multiply each element of R1R1 by 1313 to make the entry at 1,11,1 a 11.
[3353030375001100]⎡⎢
⎢⎣3353030375001100⎤⎥
⎥⎦
Step 3.3.2.1.2
Simplify R1R1.
[1530075001100]⎡⎢
⎢⎣1530075001100⎤⎥
⎥⎦
[1530075001100]⎡⎢
⎢⎣1530075001100⎤⎥
⎥⎦
Step 3.3.2.2
Perform the row operation R2=R2-7R1R2=R2−7R1 to make the entry at 2,12,1 a 00.
Step 3.3.2.2.1
Perform the row operation R2=R2-7R1R2=R2−7R1 to make the entry at 2,12,1 a 00.
[153007-7⋅15-7(53)0-7⋅00-7⋅01100]⎡⎢
⎢
⎢
⎢⎣153007−7⋅15−7(53)0−7⋅00−7⋅01100⎤⎥
⎥
⎥
⎥⎦
Step 3.3.2.2.2
Simplify R2R2.
[153000-203001100]⎡⎢
⎢
⎢⎣153000−203001100⎤⎥
⎥
⎥⎦
[153000-203001100]⎡⎢
⎢
⎢⎣153000−203001100⎤⎥
⎥
⎥⎦
Step 3.3.2.3
Perform the row operation R3=R3-R1R3=R3−R1 to make the entry at 3,13,1 a 00.
Step 3.3.2.3.1
Perform the row operation R3=R3-R1R3=R3−R1 to make the entry at 3,13,1 a 00.
[153000-203001-11-530-00-0]⎡⎢
⎢
⎢
⎢⎣153000−203001−11−530−00−0⎤⎥
⎥
⎥
⎥⎦
Step 3.3.2.3.2
Simplify R3R3.
[153000-203000-2300]⎡⎢
⎢
⎢
⎢⎣153000−203000−2300⎤⎥
⎥
⎥
⎥⎦
[153000-203000-2300]⎡⎢
⎢
⎢
⎢⎣153000−203000−2300⎤⎥
⎥
⎥
⎥⎦
Step 3.3.2.4
Multiply each element of R2R2 by -320−320 to make the entry at 2,22,2 a 11.
Step 3.3.2.4.1
Multiply each element of R2R2 by -320−320 to make the entry at 2,22,2 a 11.
[15300-320⋅0-320(-203)-320⋅0-320⋅00-2300]⎡⎢
⎢
⎢
⎢
⎢⎣15300−320⋅0−320(−203)−320⋅0−320⋅00−2300⎤⎥
⎥
⎥
⎥
⎥⎦
Step 3.3.2.4.2
Simplify R2R2.
[1530001000-2300]⎡⎢
⎢
⎢⎣1530001000−2300⎤⎥
⎥
⎥⎦
[1530001000-2300]⎡⎢
⎢
⎢⎣1530001000−2300⎤⎥
⎥
⎥⎦
Step 3.3.2.5
Perform the row operation R3=R3+23R2R3=R3+23R2 to make the entry at 3,23,2 a 00.
Step 3.3.2.5.1
Perform the row operation R3=R3+23R2R3=R3+23R2 to make the entry at 3,23,2 a 00.
[1530001000+23⋅0-23+23⋅10+23⋅00+23⋅0]⎡⎢
⎢
⎢⎣1530001000+23⋅0−23+23⋅10+23⋅00+23⋅0⎤⎥
⎥
⎥⎦
Step 3.3.2.5.2
Simplify R3R3.
[1530001000000]⎡⎢
⎢⎣1530001000000⎤⎥
⎥⎦
[1530001000000]⎡⎢
⎢⎣1530001000000⎤⎥
⎥⎦
Step 3.3.2.6
Perform the row operation R1=R1-53R2R1=R1−53R2 to make the entry at 1,21,2 a 00.
Step 3.3.2.6.1
Perform the row operation R1=R1-53R2R1=R1−53R2 to make the entry at 1,21,2 a 00.
[1-53⋅053-53⋅10-53⋅00-53⋅001000000]⎡⎢
⎢⎣1−53⋅053−53⋅10−53⋅00−53⋅001000000⎤⎥
⎥⎦
Step 3.3.2.6.2
Simplify R1R1.
[100001000000]⎡⎢
⎢⎣100001000000⎤⎥
⎥⎦
[100001000000]⎡⎢
⎢⎣100001000000⎤⎥
⎥⎦
[100001000000]⎡⎢
⎢⎣100001000000⎤⎥
⎥⎦
Step 3.3.3
Use the result matrix to declare the final solution to the system of equations.
x=0x=0
y=0y=0
0=00=0
Step 3.3.4
Write a solution vector by solving in terms of the free variables in each row.
[xyz]=[00z]⎡⎢⎣xyz⎤⎥⎦=⎡⎢⎣00z⎤⎥⎦
Step 3.3.5
Write the solution as a linear combination of vectors.
[xyz]=z[001]⎡⎢⎣xyz⎤⎥⎦=z⎡⎢⎣001⎤⎥⎦
Step 3.3.6
Write as a solution set.
{z[001]|z∈R}⎧⎪⎨⎪⎩z⎡⎢⎣001⎤⎥⎦∣∣
∣∣z∈R⎫⎪⎬⎪⎭
Step 3.3.7
The solution is the set of vectors created from the free variables of the system.
{[001]}⎧⎪⎨⎪⎩⎡⎢⎣001⎤⎥⎦⎫⎪⎬⎪⎭
{[001]}⎧⎪⎨⎪⎩⎡⎢⎣001⎤⎥⎦⎫⎪⎬⎪⎭
{[001]}⎧⎪⎨⎪⎩⎡⎢⎣001⎤⎥⎦⎫⎪⎬⎪⎭
Step 4
Step 4.1
Substitute the known values into the formula.
N([350750110]-10[100010001])N⎛⎜⎝⎡⎢⎣350750110⎤⎥⎦−10⎡⎢⎣100010001⎤⎥⎦⎞⎟⎠
Step 4.2
Simplify.
Step 4.2.1
Simplify each term.
Step 4.2.1.1
Multiply -10−10 by each element of the matrix.
[350750110]+[-10⋅1-10⋅0-10⋅0-10⋅0-10⋅1-10⋅0-10⋅0-10⋅0-10⋅1]⎡⎢⎣350750110⎤⎥⎦+⎡⎢⎣−10⋅1−10⋅0−10⋅0−10⋅0−10⋅1−10⋅0−10⋅0−10⋅0−10⋅1⎤⎥⎦
Step 4.2.1.2
Simplify each element in the matrix.
Step 4.2.1.2.1
Multiply -10−10 by 11.
[350750110]+[-10-10⋅0-10⋅0-10⋅0-10⋅1-10⋅0-10⋅0-10⋅0-10⋅1]⎡⎢⎣350750110⎤⎥⎦+⎡⎢⎣−10−10⋅0−10⋅0−10⋅0−10⋅1−10⋅0−10⋅0−10⋅0−10⋅1⎤⎥⎦
Step 4.2.1.2.2
Multiply -10−10 by 00.
[350750110]+[-100-10⋅0-10⋅0-10⋅1-10⋅0-10⋅0-10⋅0-10⋅1]⎡⎢⎣350750110⎤⎥⎦+⎡⎢⎣−100−10⋅0−10⋅0−10⋅1−10⋅0−10⋅0−10⋅0−10⋅1⎤⎥⎦
Step 4.2.1.2.3
Multiply -10−10 by 00.
[350750110]+[-1000-10⋅0-10⋅1-10⋅0-10⋅0-10⋅0-10⋅1]⎡⎢⎣350750110⎤⎥⎦+⎡⎢⎣−1000−10⋅0−10⋅1−10⋅0−10⋅0−10⋅0−10⋅1⎤⎥⎦
Step 4.2.1.2.4
Multiply -10−10 by 00.
[350750110]+[-10000-10⋅1-10⋅0-10⋅0-10⋅0-10⋅1]⎡⎢⎣350750110⎤⎥⎦+⎡⎢⎣−10000−10⋅1−10⋅0−10⋅0−10⋅0−10⋅1⎤⎥⎦
Step 4.2.1.2.5
Multiply -10−10 by 11.
[350750110]+[-10000-10-10⋅0-10⋅0-10⋅0-10⋅1]⎡⎢⎣350750110⎤⎥⎦+⎡⎢⎣−10000−10−10⋅0−10⋅0−10⋅0−10⋅1⎤⎥⎦
Step 4.2.1.2.6
Multiply -10−10 by 00.
[350750110]+[-10000-100-10⋅0-10⋅0-10⋅1]⎡⎢⎣350750110⎤⎥⎦+⎡⎢⎣−10000−100−10⋅0−10⋅0−10⋅1⎤⎥⎦
Step 4.2.1.2.7
Multiply -10−10 by 00.
[350750110]+[-10000-1000-10⋅0-10⋅1]⎡⎢⎣350750110⎤⎥⎦+⎡⎢⎣−10000−1000−10⋅0−10⋅1⎤⎥⎦
Step 4.2.1.2.8
Multiply -10−10 by 00.
[350750110]+[-10000-10000-10⋅1]⎡⎢⎣350750110⎤⎥⎦+⎡⎢⎣−10000−10000−10⋅1⎤⎥⎦
Step 4.2.1.2.9
Multiply -10−10 by 11.
[350750110]+[-10000-10000-10]⎡⎢⎣350750110⎤⎥⎦+⎡⎢⎣−10000−10000−10⎤⎥⎦
[350750110]+[-10000-10000-10]⎡⎢⎣350750110⎤⎥⎦+⎡⎢⎣−10000−10000−10⎤⎥⎦
[350750110]+[-10000-10000-10]⎡⎢⎣350750110⎤⎥⎦+⎡⎢⎣−10000−10000−10⎤⎥⎦
Step 4.2.2
Add the corresponding elements.
[3-105+00+07+05-100+01+01+00-10]⎡⎢⎣3−105+00+07+05−100+01+01+00−10⎤⎥⎦
Step 4.2.3
Simplify each element.
Step 4.2.3.1
Subtract 1010 from 33.
[-75+00+07+05-100+01+01+00-10]⎡⎢⎣−75+00+07+05−100+01+01+00−10⎤⎥⎦
Step 4.2.3.2
Add 55 and 00.
[-750+07+05-100+01+01+00-10]⎡⎢⎣−750+07+05−100+01+01+00−10⎤⎥⎦
Step 4.2.3.3
Add 00 and 00.
[-7507+05-100+01+01+00-10]⎡⎢⎣−7507+05−100+01+01+00−10⎤⎥⎦
Step 4.2.3.4
Add 77 and 00.
[-75075-100+01+01+00-10]⎡⎢⎣−75075−100+01+01+00−10⎤⎥⎦
Step 4.2.3.5
Subtract 1010 from 55.
[-7507-50+01+01+00-10]⎡⎢⎣−7507−50+01+01+00−10⎤⎥⎦
Step 4.2.3.6
Add 00 and 00.
[-7507-501+01+00-10]⎡⎢⎣−7507−501+01+00−10⎤⎥⎦
Step 4.2.3.7
Add 11 and 00.
[-7507-5011+00-10]⎡⎢⎣−7507−5011+00−10⎤⎥⎦
Step 4.2.3.8
Add 11 and 00.
[-7507-50110-10]⎡⎢⎣−7507−50110−10⎤⎥⎦
Step 4.2.3.9
Subtract 1010 from 00.
[-7507-5011-10]⎡⎢⎣−7507−5011−10⎤⎥⎦
[-7507-5011-10]⎡⎢⎣−7507−5011−10⎤⎥⎦
[-7507-5011-10]⎡⎢⎣−7507−5011−10⎤⎥⎦
Step 4.3
Find the null space when λ=10λ=10.
Step 4.3.1
Write as an augmented matrix for Ax=0Ax=0.
[-75007-50011-100]⎡⎢
⎢⎣−75007−50011−100⎤⎥
⎥⎦
Step 4.3.2
Find the reduced row echelon form.
Step 4.3.2.1
Multiply each element of R1R1 by -17−17 to make the entry at 1,11,1 a 11.
Step 4.3.2.1.1
Multiply each element of R1R1 by -17−17 to make the entry at 1,11,1 a 11.
[-17⋅-7-17⋅5-17⋅0-17⋅07-50011-100]⎡⎢
⎢⎣−17⋅−7−17⋅5−17⋅0−17⋅07−50011−100⎤⎥
⎥⎦
Step 4.3.2.1.2
Simplify R1R1.
[1-57007-50011-100]⎡⎢
⎢⎣1−57007−50011−100⎤⎥
⎥⎦
[1-57007-50011-100]⎡⎢
⎢⎣1−57007−50011−100⎤⎥
⎥⎦
Step 4.3.2.2
Perform the row operation R2=R2-7R1R2=R2−7R1 to make the entry at 2,12,1 a 00.
Step 4.3.2.2.1
Perform the row operation R2=R2-7R1R2=R2−7R1 to make the entry at 2,12,1 a 00.
[1-57007-7⋅1-5-7(-57)0-7⋅00-7⋅011-100]⎡⎢
⎢
⎢
⎢⎣1−57007−7⋅1−5−7(−57)0−7⋅00−7⋅011−100⎤⎥
⎥
⎥
⎥⎦
Step 4.3.2.2.2
Simplify R2R2.
[1-5700000011-100]⎡⎢
⎢⎣1−5700000011−100⎤⎥
⎥⎦
[1-5700000011-100]⎡⎢
⎢⎣1−5700000011−100⎤⎥
⎥⎦
Step 4.3.2.3
Perform the row operation R3=R3-R1R3=R3−R1 to make the entry at 3,13,1 a 00.
Step 4.3.2.3.1
Perform the row operation R3=R3-R1R3=R3−R1 to make the entry at 3,13,1 a 00.
[1-570000001-11+57-10-00-0]⎡⎢
⎢
⎢⎣1−570000001−11+57−10−00−0⎤⎥
⎥
⎥⎦
Step 4.3.2.3.2
Simplify R3R3.
[1-570000000127-100]⎡⎢
⎢
⎢⎣1−570000000127−100⎤⎥
⎥
⎥⎦
[1-570000000127-100]⎡⎢
⎢
⎢⎣1−570000000127−100⎤⎥
⎥
⎥⎦
Step 4.3.2.4
Swap R3R3 with R2R2 to put a nonzero entry at 2,22,2.
[1-57000127-1000000]⎡⎢
⎢
⎢⎣1−57000127−1000000⎤⎥
⎥
⎥⎦
Step 4.3.2.5
Multiply each element of R2R2 by 712712 to make the entry at 2,22,2 a 11.
Step 4.3.2.5.1
Multiply each element of R2R2 by 712712 to make the entry at 2,22,2 a 11.
[1-5700712⋅0712⋅127712⋅-10712⋅00000]⎡⎢
⎢
⎢⎣1−5700712⋅0712⋅127712⋅−10712⋅00000⎤⎥
⎥
⎥⎦
Step 4.3.2.5.2
Simplify R2R2.
[1-570001-35600000]⎡⎢
⎢
⎢⎣1−570001−35600000⎤⎥
⎥
⎥⎦
[1-570001-35600000]⎡⎢
⎢
⎢⎣1−570001−35600000⎤⎥
⎥
⎥⎦
Step 4.3.2.6
Perform the row operation R1=R1+57R2R1=R1+57R2 to make the entry at 1,21,2 a 00.
Step 4.3.2.6.1
Perform the row operation R1=R1+57R2R1=R1+57R2 to make the entry at 1,21,2 a 00.
[1+57⋅0-57+57⋅10+57(-356)0+57⋅001-35600000]⎡⎢
⎢
⎢
⎢⎣1+57⋅0−57+57⋅10+57(−356)0+57⋅001−35600000⎤⎥
⎥
⎥
⎥⎦
Step 4.3.2.6.2
Simplify R1R1.
[10-256001-35600000]⎡⎢
⎢
⎢⎣10−256001−35600000⎤⎥
⎥
⎥⎦
[10-256001-35600000]⎡⎢
⎢
⎢⎣10−256001−35600000⎤⎥
⎥
⎥⎦
[10-256001-35600000]⎡⎢
⎢
⎢⎣10−256001−35600000⎤⎥
⎥
⎥⎦
Step 4.3.3
Use the result matrix to declare the final solution to the system of equations.
x-256z=0x−256z=0
y-356z=0y−356z=0
0=00=0
Step 4.3.4
Write a solution vector by solving in terms of the free variables in each row.
[xyz]=[25z635z6z]⎡⎢⎣xyz⎤⎥⎦=⎡⎢
⎢⎣25z635z6z⎤⎥
⎥⎦
Step 4.3.5
Write the solution as a linear combination of vectors.
[xyz]=z[2563561]⎡⎢⎣xyz⎤⎥⎦=z⎡⎢
⎢⎣2563561⎤⎥
⎥⎦
Step 4.3.6
Write as a solution set.
{z[2563561]|z∈R}⎧⎪
⎪⎨⎪
⎪⎩z⎡⎢
⎢⎣2563561⎤⎥
⎥⎦∣∣
∣
∣∣z∈R⎫⎪
⎪⎬⎪
⎪⎭
Step 4.3.7
The solution is the set of vectors created from the free variables of the system.
{[2563561]}⎧⎪
⎪⎨⎪
⎪⎩⎡⎢
⎢⎣2563561⎤⎥
⎥⎦⎫⎪
⎪⎬⎪
⎪⎭
{[2563561]}⎧⎪
⎪⎨⎪
⎪⎩⎡⎢
⎢⎣2563561⎤⎥
⎥⎦⎫⎪
⎪⎬⎪
⎪⎭
{[2563561]}⎧⎪
⎪⎨⎪
⎪⎩⎡⎢
⎢⎣2563561⎤⎥
⎥⎦⎫⎪
⎪⎬⎪
⎪⎭
Step 5
Step 5.1
Substitute the known values into the formula.
N([350750110]+2[100010001])N⎛⎜⎝⎡⎢⎣350750110⎤⎥⎦+2⎡⎢⎣100010001⎤⎥⎦⎞⎟⎠
Step 5.2
Simplify.
Step 5.2.1
Simplify each term.
Step 5.2.1.1
Multiply 22 by each element of the matrix.
[350750110]+[2⋅12⋅02⋅02⋅02⋅12⋅02⋅02⋅02⋅1]⎡⎢⎣350750110⎤⎥⎦+⎡⎢⎣2⋅12⋅02⋅02⋅02⋅12⋅02⋅02⋅02⋅1⎤⎥⎦
Step 5.2.1.2
Simplify each element in the matrix.
Step 5.2.1.2.1
Multiply 22 by 11.
[350750110]+[22⋅02⋅02⋅02⋅12⋅02⋅02⋅02⋅1]⎡⎢⎣350750110⎤⎥⎦+⎡⎢⎣22⋅02⋅02⋅02⋅12⋅02⋅02⋅02⋅1⎤⎥⎦
Step 5.2.1.2.2
Multiply 22 by 00.
[350750110]+[202⋅02⋅02⋅12⋅02⋅02⋅02⋅1]⎡⎢⎣350750110⎤⎥⎦+⎡⎢⎣202⋅02⋅02⋅12⋅02⋅02⋅02⋅1⎤⎥⎦
Step 5.2.1.2.3
Multiply 22 by 00.
[350750110]+[2002⋅02⋅12⋅02⋅02⋅02⋅1]⎡⎢⎣350750110⎤⎥⎦+⎡⎢⎣2002⋅02⋅12⋅02⋅02⋅02⋅1⎤⎥⎦
Step 5.2.1.2.4
Multiply 22 by 00.
[350750110]+[20002⋅12⋅02⋅02⋅02⋅1]⎡⎢⎣350750110⎤⎥⎦+⎡⎢⎣20002⋅12⋅02⋅02⋅02⋅1⎤⎥⎦
Step 5.2.1.2.5
Multiply 22 by 11.
[350750110]+[200022⋅02⋅02⋅02⋅1]⎡⎢⎣350750110⎤⎥⎦+⎡⎢⎣200022⋅02⋅02⋅02⋅1⎤⎥⎦
Step 5.2.1.2.6
Multiply 22 by 00.
[350750110]+[2000202⋅02⋅02⋅1]⎡⎢⎣350750110⎤⎥⎦+⎡⎢⎣2000202⋅02⋅02⋅1⎤⎥⎦
Step 5.2.1.2.7
Multiply 22 by 00.
[350750110]+[20002002⋅02⋅1]⎡⎢⎣350750110⎤⎥⎦+⎡⎢⎣20002002⋅02⋅1⎤⎥⎦
Step 5.2.1.2.8
Multiply 22 by 00.
[350750110]+[200020002⋅1]⎡⎢⎣350750110⎤⎥⎦+⎡⎢⎣200020002⋅1⎤⎥⎦
Step 5.2.1.2.9
Multiply 22 by 11.
[350750110]+[200020002]⎡⎢⎣350750110⎤⎥⎦+⎡⎢⎣200020002⎤⎥⎦
[350750110]+[200020002]⎡⎢⎣350750110⎤⎥⎦+⎡⎢⎣200020002⎤⎥⎦
[350750110]+[200020002]⎡⎢⎣350750110⎤⎥⎦+⎡⎢⎣200020002⎤⎥⎦
Step 5.2.2
Add the corresponding elements.
[3+25+00+07+05+20+01+01+00+2]⎡⎢⎣3+25+00+07+05+20+01+01+00+2⎤⎥⎦
Step 5.2.3
Simplify each element.
Step 5.2.3.1
Add 33 and 22.
[55+00+07+05+20+01+01+00+2]⎡⎢⎣55+00+07+05+20+01+01+00+2⎤⎥⎦
Step 5.2.3.2
Add 55 and 00.
[550+07+05+20+01+01+00+2]⎡⎢⎣550+07+05+20+01+01+00+2⎤⎥⎦
Step 5.2.3.3
Add 00 and 00.
[5507+05+20+01+01+00+2]⎡⎢⎣5507+05+20+01+01+00+2⎤⎥⎦
Step 5.2.3.4
Add 77 and 00.
[55075+20+01+01+00+2]⎡⎢⎣55075+20+01+01+00+2⎤⎥⎦
Step 5.2.3.5
Add 55 and 22.
[550770+01+01+00+2]⎡⎢⎣550770+01+01+00+2⎤⎥⎦
Step 5.2.3.6
Add 00 and 00.
[5507701+01+00+2]⎡⎢⎣5507701+01+00+2⎤⎥⎦
Step 5.2.3.7
Add 11 and 00.
[55077011+00+2]⎡⎢⎣55077011+00+2⎤⎥⎦
Step 5.2.3.8
Add 11 and 00.
[550770110+2]⎡⎢⎣550770110+2⎤⎥⎦
Step 5.2.3.9
Add 00 and 22.
[550770112]⎡⎢⎣550770112⎤⎥⎦
[550770112]⎡⎢⎣550770112⎤⎥⎦
[550770112]⎡⎢⎣550770112⎤⎥⎦
Step 5.3
Find the null space when λ=-2λ=−2.
Step 5.3.1
Write as an augmented matrix for Ax=0Ax=0.
[550077001120]⎡⎢
⎢⎣550077001120⎤⎥
⎥⎦
Step 5.3.2
Find the reduced row echelon form.
Step 5.3.2.1
Multiply each element of R1R1 by 1515 to make the entry at 1,11,1 a 11.
Step 5.3.2.1.1
Multiply each element of R1R1 by 1515 to make the entry at 1,11,1 a 11.
[5555050577001120]⎡⎢
⎢⎣5555050577001120⎤⎥
⎥⎦
Step 5.3.2.1.2
Simplify R1R1.
[110077001120]⎡⎢
⎢⎣110077001120⎤⎥
⎥⎦
[110077001120]⎡⎢
⎢⎣110077001120⎤⎥
⎥⎦
Step 5.3.2.2
Perform the row operation R2=R2-7R1R2=R2−7R1 to make the entry at 2,12,1 a 00.
Step 5.3.2.2.1
Perform the row operation R2=R2-7R1R2=R2−7R1 to make the entry at 2,12,1 a 00.
[11007-7⋅17-7⋅10-7⋅00-7⋅01120]⎡⎢
⎢⎣11007−7⋅17−7⋅10−7⋅00−7⋅01120⎤⎥
⎥⎦
Step 5.3.2.2.2
Simplify R2R2.
[110000001120]⎡⎢
⎢⎣110000001120⎤⎥
⎥⎦
[110000001120]⎡⎢
⎢⎣110000001120⎤⎥
⎥⎦
Step 5.3.2.3
Perform the row operation R3=R3-R1R3=R3−R1 to make the entry at 3,13,1 a 00.
Step 5.3.2.3.1
Perform the row operation R3=R3-R1R3=R3−R1 to make the entry at 3,13,1 a 00.
[110000001-11-12-00-0]⎡⎢
⎢⎣110000001−11−12−00−0⎤⎥
⎥⎦
Step 5.3.2.3.2
Simplify R3R3.
[110000000020]⎡⎢
⎢⎣110000000020⎤⎥
⎥⎦
[110000000020]⎡⎢
⎢⎣110000000020⎤⎥
⎥⎦
Step 5.3.2.4
Swap R3 with R2 to put a nonzero entry at 2,3.
[110000200000]
Step 5.3.2.5
Multiply each element of R2 by 12 to make the entry at 2,3 a 1.
Step 5.3.2.5.1
Multiply each element of R2 by 12 to make the entry at 2,3 a 1.
[1100020222020000]
Step 5.3.2.5.2
Simplify R2.
[110000100000]
[110000100000]
[110000100000]
Step 5.3.3
Use the result matrix to declare the final solution to the system of equations.
x+y=0
z=0
0=0
Step 5.3.4
Write a solution vector by solving in terms of the free variables in each row.
[xyz]=[-yy0]
Step 5.3.5
Write the solution as a linear combination of vectors.
[xyz]=y[-110]
Step 5.3.6
Write as a solution set.
{y[-110]|y∈R}
Step 5.3.7
The solution is the set of vectors created from the free variables of the system.
{[-110]}
{[-110]}
{[-110]}
Step 6
The eigenspace of A is the list of the vector space for each eigenvalue.
{[001],[2563561],[-110]}
