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线性代数 示例
25x+30y+30z=147525x+30y+30z=1475 , 50x+30y+20z=99050x+30y+20z=990 , 75x+30y+20z=81075x+30y+20z=810
解题步骤 1
从方程组中求 AX=BAX=B。
[253030503020753020]⋅[xyz]=[1475990810]⎡⎢⎣253030503020753020⎤⎥⎦⋅⎡⎢⎣xyz⎤⎥⎦=⎡⎢⎣1475990810⎤⎥⎦
解题步骤 2
解题步骤 2.1
Find the determinant.
解题步骤 2.1.1
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 row 11 by its cofactor and add.
解题步骤 2.1.1.1
Consider the corresponding sign chart.
|+-+-+-+-+|∣∣
∣∣+−+−+−+−+∣∣
∣∣
解题步骤 2.1.1.2
The cofactor is the minor with the sign changed if the indices match a -− position on the sign chart.
解题步骤 2.1.1.3
The minor for a11a11 is the determinant with row 11 and column 11 deleted.
|30203020|∣∣∣30203020∣∣∣
解题步骤 2.1.1.4
Multiply element a11a11 by its cofactor.
25|30203020|25∣∣∣30203020∣∣∣
解题步骤 2.1.1.5
The minor for a12a12 is the determinant with row 11 and column 22 deleted.
|50207520|∣∣∣50207520∣∣∣
解题步骤 2.1.1.6
Multiply element a12a12 by its cofactor.
-30|50207520|−30∣∣∣50207520∣∣∣
解题步骤 2.1.1.7
The minor for a13a13 is the determinant with row 11 and column 33 deleted.
|50307530|∣∣∣50307530∣∣∣
解题步骤 2.1.1.8
Multiply element a13a13 by its cofactor.
30|50307530|30∣∣∣50307530∣∣∣
解题步骤 2.1.1.9
Add the terms together.
25|30203020|-30|50207520|+30|50307530|25∣∣∣30203020∣∣∣−30∣∣∣50207520∣∣∣+30∣∣∣50307530∣∣∣
25|30203020|-30|50207520|+30|50307530|25∣∣∣30203020∣∣∣−30∣∣∣50207520∣∣∣+30∣∣∣50307530∣∣∣
解题步骤 2.1.2
计算 |30203020|∣∣∣30203020∣∣∣。
解题步骤 2.1.2.1
可以使用公式 |abcd|=ad-cb∣∣∣abcd∣∣∣=ad−cb 求 2×22×2 矩阵的行列式。
25(30⋅20-30⋅20)-30|50207520|+30|50307530|25(30⋅20−30⋅20)−30∣∣∣50207520∣∣∣+30∣∣∣50307530∣∣∣
解题步骤 2.1.2.2
化简行列式。
解题步骤 2.1.2.2.1
化简每一项。
解题步骤 2.1.2.2.1.1
将 3030 乘以 2020。
25(600-30⋅20)-30|50207520|+30|50307530|25(600−30⋅20)−30∣∣∣50207520∣∣∣+30∣∣∣50307530∣∣∣
解题步骤 2.1.2.2.1.2
将 -30−30 乘以 2020。
25(600-600)-30|50207520|+30|50307530|25(600−600)−30∣∣∣50207520∣∣∣+30∣∣∣50307530∣∣∣
25(600-600)-30|50207520|+30|50307530|25(600−600)−30∣∣∣50207520∣∣∣+30∣∣∣50307530∣∣∣
解题步骤 2.1.2.2.2
从 600600 中减去 600600。
25⋅0-30|50207520|+30|50307530|25⋅0−30∣∣∣50207520∣∣∣+30∣∣∣50307530∣∣∣
25⋅0-30|50207520|+30|50307530|25⋅0−30∣∣∣50207520∣∣∣+30∣∣∣50307530∣∣∣
25⋅0-30|50207520|+30|50307530|25⋅0−30∣∣∣50207520∣∣∣+30∣∣∣50307530∣∣∣
解题步骤 2.1.3
计算 |50207520|∣∣∣50207520∣∣∣。
解题步骤 2.1.3.1
可以使用公式 |abcd|=ad-cb∣∣∣abcd∣∣∣=ad−cb 求 2×22×2 矩阵的行列式。
25⋅0-30(50⋅20-75⋅20)+30|50307530|25⋅0−30(50⋅20−75⋅20)+30∣∣∣50307530∣∣∣
解题步骤 2.1.3.2
化简行列式。
解题步骤 2.1.3.2.1
化简每一项。
解题步骤 2.1.3.2.1.1
将 5050 乘以 2020。
25⋅0-30(1000-75⋅20)+30|50307530|25⋅0−30(1000−75⋅20)+30∣∣∣50307530∣∣∣
解题步骤 2.1.3.2.1.2
将 -75−75 乘以 2020。
25⋅0-30(1000-1500)+30|50307530|25⋅0−30(1000−1500)+30∣∣∣50307530∣∣∣
25⋅0-30(1000-1500)+30|50307530|25⋅0−30(1000−1500)+30∣∣∣50307530∣∣∣
解题步骤 2.1.3.2.2
从 10001000 中减去 15001500。
25⋅0-30⋅-500+30|50307530|25⋅0−30⋅−500+30∣∣∣50307530∣∣∣
25⋅0-30⋅-500+30|50307530|25⋅0−30⋅−500+30∣∣∣50307530∣∣∣
25⋅0-30⋅-500+30|50307530|25⋅0−30⋅−500+30∣∣∣50307530∣∣∣
解题步骤 2.1.4
计算 |50307530|∣∣∣50307530∣∣∣。
解题步骤 2.1.4.1
可以使用公式 |abcd|=ad-cb∣∣∣abcd∣∣∣=ad−cb 求 2×22×2 矩阵的行列式。
25⋅0-30⋅-500+30(50⋅30-75⋅30)25⋅0−30⋅−500+30(50⋅30−75⋅30)
解题步骤 2.1.4.2
化简行列式。
解题步骤 2.1.4.2.1
化简每一项。
解题步骤 2.1.4.2.1.1
将 5050 乘以 3030。
25⋅0-30⋅-500+30(1500-75⋅30)25⋅0−30⋅−500+30(1500−75⋅30)
解题步骤 2.1.4.2.1.2
将 -75−75 乘以 3030。
25⋅0-30⋅-500+30(1500-2250)25⋅0−30⋅−500+30(1500−2250)
25⋅0-30⋅-500+30(1500-2250)25⋅0−30⋅−500+30(1500−2250)
解题步骤 2.1.4.2.2
从 15001500 中减去 22502250。
25⋅0-30⋅-500+30⋅-75025⋅0−30⋅−500+30⋅−750
25⋅0-30⋅-500+30⋅-75025⋅0−30⋅−500+30⋅−750
25⋅0-30⋅-500+30⋅-75025⋅0−30⋅−500+30⋅−750
解题步骤 2.1.5
化简行列式。
解题步骤 2.1.5.1
化简每一项。
解题步骤 2.1.5.1.1
将 2525 乘以 00。
0-30⋅-500+30⋅-7500−30⋅−500+30⋅−750
解题步骤 2.1.5.1.2
将 -30−30 乘以 -500−500。
0+15000+30⋅-7500+15000+30⋅−750
解题步骤 2.1.5.1.3
将 3030 乘以 -750−750。
0+15000-225000+15000−22500
0+15000-225000+15000−22500
解题步骤 2.1.5.2
将 00 和 1500015000 相加。
15000-2250015000−22500
解题步骤 2.1.5.3
从 1500015000 中减去 2250022500。
-7500−7500
-7500−7500
-7500−7500
解题步骤 2.2
Since the determinant is non-zero, the inverse exists.
解题步骤 2.3
Set up a 3×63×6 matrix where the left half is the original matrix and the right half is its identity matrix.
[253030100503020010753020001]⎡⎢⎣253030100503020010753020001⎤⎥⎦
解题步骤 2.4
求行简化阶梯形矩阵。
解题步骤 2.4.1
Multiply each element of R1R1 by 125125 to make the entry at 1,11,1 a 11.
解题步骤 2.4.1.1
Multiply each element of R1R1 by 125125 to make the entry at 1,11,1 a 11.
[252530253025125025025503020010753020001]⎡⎢
⎢⎣252530253025125025025503020010753020001⎤⎥
⎥⎦
解题步骤 2.4.1.2
化简 R1R1。
[1656512500503020010753020001]⎡⎢
⎢⎣1656512500503020010753020001⎤⎥
⎥⎦
[1656512500503020010753020001]⎡⎢
⎢⎣1656512500503020010753020001⎤⎥
⎥⎦
解题步骤 2.4.2
Perform the row operation R2=R2-50R1R2=R2−50R1 to make the entry at 2,12,1 a 00.
解题步骤 2.4.2.1
Perform the row operation R2=R2-50R1R2=R2−50R1 to make the entry at 2,12,1 a 00.
[165651250050-50⋅130-50(65)20-50(65)0-50(125)1-50⋅00-50⋅0753020001]⎡⎢
⎢
⎢⎣165651250050−50⋅130−50(65)20−50(65)0−50(125)1−50⋅00−50⋅0753020001⎤⎥
⎥
⎥⎦
解题步骤 2.4.2.2
化简 R2R2。
[16565125000-30-40-210753020001]⎡⎢
⎢⎣16565125000−30−40−210753020001⎤⎥
⎥⎦
[16565125000-30-40-210753020001]⎡⎢
⎢⎣16565125000−30−40−210753020001⎤⎥
⎥⎦
解题步骤 2.4.3
Perform the row operation R3=R3-75R1R3=R3−75R1 to make the entry at 3,13,1 a 00.
解题步骤 2.4.3.1
Perform the row operation R3=R3-75R1R3=R3−75R1 to make the entry at 3,13,1 a 00.
[16565125000-30-40-21075-75⋅130-75(65)20-75(65)0-75(125)0-75⋅01-75⋅0]⎡⎢
⎢
⎢⎣16565125000−30−40−21075−75⋅130−75(65)20−75(65)0−75(125)0−75⋅01−75⋅0⎤⎥
⎥
⎥⎦
解题步骤 2.4.3.2
化简 R3R3。
[16565125000-30-40-2100-60-70-301]⎡⎢
⎢⎣16565125000−30−40−2100−60−70−301⎤⎥
⎥⎦
[16565125000-30-40-2100-60-70-301]⎡⎢
⎢⎣16565125000−30−40−2100−60−70−301⎤⎥
⎥⎦
解题步骤 2.4.4
Multiply each element of R2R2 by -130−130 to make the entry at 2,22,2 a 11.
解题步骤 2.4.4.1
Multiply each element of R2R2 by -130−130 to make the entry at 2,22,2 a 11.
[1656512500-130⋅0-130⋅-30-130⋅-40-130⋅-2-130⋅1-130⋅00-60-70-301]⎡⎢
⎢⎣1656512500−130⋅0−130⋅−30−130⋅−40−130⋅−2−130⋅1−130⋅00−60−70−301⎤⎥
⎥⎦
解题步骤 2.4.4.2
化简 R2R2。
[16565125000143115-13000-60-70-301]⎡⎢
⎢⎣16565125000143115−13000−60−70−301⎤⎥
⎥⎦
[16565125000143115-13000-60-70-301]⎡⎢
⎢⎣16565125000143115−13000−60−70−301⎤⎥
⎥⎦
解题步骤 2.4.5
Perform the row operation R3=R3+60R2R3=R3+60R2 to make the entry at 3,23,2 a 00.
解题步骤 2.4.5.1
Perform the row operation R3=R3+60R2R3=R3+60R2 to make the entry at 3,23,2 a 00.
[16565125000143115-13000+60⋅0-60+60⋅1-70+60(43)-3+60(115)0+60(-130)1+60⋅0]⎡⎢
⎢
⎢
⎢⎣16565125000143115−13000+60⋅0−60+60⋅1−70+60(43)−3+60(115)0+60(−130)1+60⋅0⎤⎥
⎥
⎥
⎥⎦
解题步骤 2.4.5.2
化简 R3R3。
[16565125000143115-130000101-21]⎡⎢
⎢⎣16565125000143115−130000101−21⎤⎥
⎥⎦
[16565125000143115-130000101-21]⎡⎢
⎢⎣16565125000143115−130000101−21⎤⎥
⎥⎦
解题步骤 2.4.6
Multiply each element of R3R3 by 110110 to make the entry at 3,33,3 a 11.
解题步骤 2.4.6.1
Multiply each element of R3R3 by 110110 to make the entry at 3,33,3 a 11.
[16565125000143115-13000100101010110-210110]⎡⎢
⎢
⎢⎣16565125000143115−13000100101010110−210110⎤⎥
⎥
⎥⎦
解题步骤 2.4.6.2
化简 R3R3。
[16565125000143115-1300001110-15110]⎡⎢
⎢
⎢⎣16565125000143115−1300001110−15110⎤⎥
⎥
⎥⎦
[16565125000143115-1300001110-15110]⎡⎢
⎢
⎢⎣16565125000143115−1300001110−15110⎤⎥
⎥
⎥⎦
解题步骤 2.4.7
Perform the row operation R2=R2-43R3R2=R2−43R3 to make the entry at 2,32,3 a 00.
解题步骤 2.4.7.1
Perform the row operation R2=R2-43R3R2=R2−43R3 to make the entry at 2,32,3 a 00.
[16565125000-43⋅01-43⋅043-43⋅1115-43⋅110-130-43(-15)0-43⋅110001110-15110]⎡⎢
⎢
⎢
⎢⎣16565125000−43⋅01−43⋅043−43⋅1115−43⋅110−130−43(−15)0−43⋅110001110−15110⎤⎥
⎥
⎥
⎥⎦
解题步骤 2.4.7.2
化简 R2。
[1656512500010-115730-215001110-15110]
[1656512500010-115730-215001110-15110]
解题步骤 2.4.8
Perform the row operation R1=R1-65R3 to make the entry at 1,3 a 0.
解题步骤 2.4.8.1
Perform the row operation R1=R1-65R3 to make the entry at 1,3 a 0.
[1-65⋅065-65⋅065-65⋅1125-65⋅1100-65(-15)0-65⋅110010-115730-215001110-15110]
解题步骤 2.4.8.2
化简 R1。
[1650-225625-325010-115730-215001110-15110]
[1650-225625-325010-115730-215001110-15110]
解题步骤 2.4.9
Perform the row operation R1=R1-65R2 to make the entry at 1,2 a 0.
解题步骤 2.4.9.1
Perform the row operation R1=R1-65R2 to make the entry at 1,2 a 0.
[1-65⋅065-65⋅10-65⋅0-225-65(-115)625-65⋅730-325-65(-215)010-115730-215001110-15110]
解题步骤 2.4.9.2
化简 R1。
[1000-125125010-115730-215001110-15110]
[1000-125125010-115730-215001110-15110]
[1000-125125010-115730-215001110-15110]
解题步骤 2.5
The right half of the reduced row echelon form is the inverse.
[0-125125-115730-215110-15110]
[0-125125-115730-215110-15110]
解题步骤 3
对矩阵方程的两边同时左乘逆矩阵。
([0-125125-115730-215110-15110]⋅[253030503020753020])⋅[xyz]=[0-125125-115730-215110-15110]⋅[1475990810]
解题步骤 4
任何矩阵与其逆矩阵的乘积始终等于 1。A⋅A-1=1。
[xyz]=[0-125125-115730-215110-15110]⋅[1475990810]
解题步骤 5
解题步骤 5.1
Two matrices can be multiplied if and only if the number of columns in the first matrix is equal to the number of rows in the second matrix. In this case, the first matrix is 3×3 and the second matrix is 3×1.
解题步骤 5.2
将第一个矩阵中的每一行乘以第二个矩阵中的每一列。
[0⋅1475-125⋅990+125⋅810-115⋅1475+730⋅990-215⋅810110⋅1475-15⋅990+110⋅810]
解题步骤 5.3
通过展开所有表达式化简矩阵的每一个元素。
[-365743612]
[-365743612]
解题步骤 6
化简左右两边。
[xyz]=[-365743612]
解题步骤 7
求解。
x=-365
y=743
z=612