Finite Math Examples

Solve the Matrix Equation A[[8,-5,-4],[1,-4,4],[-6,-2,9]]B=[[-7,2,5],[9,-9,4],[5,-1,5]]
Step 1
Multiply by each element of the matrix.
Step 2
Simplify each element in the matrix.
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Step 2.1
Move to the left of .
Step 2.2
Move to the left of .
Step 2.3
Move to the left of .
Step 2.4
Multiply by .
Step 2.5
Move to the left of .
Step 2.6
Move to the left of .
Step 2.7
Move to the left of .
Step 2.8
Move to the left of .
Step 2.9
Move to the left of .
Step 3
Find the inverse of .
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Step 3.1
Rewrite.
Step 3.2
Find the determinant.
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Step 3.2.1
Choose the row or column with the most elements. If there are no elements choose any row or column. Multiply every element in row by its cofactor and add.
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Step 3.2.1.1
Consider the corresponding sign chart.
Step 3.2.1.2
The cofactor is the minor with the sign changed if the indices match a position on the sign chart.
Step 3.2.1.3
The minor for is the determinant with row and column deleted.
Step 3.2.1.4
Multiply element by its cofactor.
Step 3.2.1.5
The minor for is the determinant with row and column deleted.
Step 3.2.1.6
Multiply element by its cofactor.
Step 3.2.1.7
The minor for is the determinant with row and column deleted.
Step 3.2.1.8
Multiply element by its cofactor.
Step 3.2.1.9
Add the terms together.
Step 3.2.2
Evaluate .
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Step 3.2.2.1
The determinant of a matrix can be found using the formula .
Step 3.2.2.2
Simplify the determinant.
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Step 3.2.2.2.1
Simplify each term.
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Step 3.2.2.2.1.1
Rewrite using the commutative property of multiplication.
Step 3.2.2.2.1.2
Multiply by by adding the exponents.
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Step 3.2.2.2.1.2.1
Move .
Step 3.2.2.2.1.2.2
Multiply by .
Step 3.2.2.2.1.3
Multiply by .
Step 3.2.2.2.1.4
Multiply by by adding the exponents.
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Step 3.2.2.2.1.4.1
Move .
Step 3.2.2.2.1.4.2
Multiply by .
Step 3.2.2.2.1.5
Multiply by .
Step 3.2.2.2.1.6
Multiply by .
Step 3.2.2.2.2
Add and .
Step 3.2.3
Evaluate .
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Step 3.2.3.1
The determinant of a matrix can be found using the formula .
Step 3.2.3.2
Simplify the determinant.
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Step 3.2.3.2.1
Simplify each term.
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Step 3.2.3.2.1.1
Rewrite using the commutative property of multiplication.
Step 3.2.3.2.1.2
Multiply by by adding the exponents.
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Step 3.2.3.2.1.2.1
Move .
Step 3.2.3.2.1.2.2
Multiply by .
Step 3.2.3.2.1.3
Multiply by by adding the exponents.
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Step 3.2.3.2.1.3.1
Move .
Step 3.2.3.2.1.3.2
Multiply by .
Step 3.2.3.2.1.4
Multiply by .
Step 3.2.3.2.1.5
Multiply by .
Step 3.2.3.2.2
Add and .
Step 3.2.4
Evaluate .
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Step 3.2.4.1
The determinant of a matrix can be found using the formula .
Step 3.2.4.2
Simplify the determinant.
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Step 3.2.4.2.1
Simplify each term.
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Step 3.2.4.2.1.1
Rewrite using the commutative property of multiplication.
Step 3.2.4.2.1.2
Multiply by by adding the exponents.
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Step 3.2.4.2.1.2.1
Move .
Step 3.2.4.2.1.2.2
Multiply by .
Step 3.2.4.2.1.3
Multiply by by adding the exponents.
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Step 3.2.4.2.1.3.1
Move .
Step 3.2.4.2.1.3.2
Multiply by .
Step 3.2.4.2.1.4
Multiply by .
Step 3.2.4.2.1.5
Multiply by .
Step 3.2.4.2.2
Subtract from .
Step 3.2.5
Simplify the determinant.
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Step 3.2.5.1
Simplify each term.
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Step 3.2.5.1.1
Rewrite using the commutative property of multiplication.
Step 3.2.5.1.2
Multiply by by adding the exponents.
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Step 3.2.5.1.2.1
Move .
Step 3.2.5.1.2.2
Multiply by .
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Step 3.2.5.1.2.2.1
Raise to the power of .
Step 3.2.5.1.2.2.2
Use the power rule to combine exponents.
Step 3.2.5.1.2.3
Add and .
Step 3.2.5.1.3
Multiply by .
Step 3.2.5.1.4
Rewrite using the commutative property of multiplication.
Step 3.2.5.1.5
Multiply by by adding the exponents.
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Step 3.2.5.1.5.1
Move .
Step 3.2.5.1.5.2
Multiply by .
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Step 3.2.5.1.5.2.1
Raise to the power of .
Step 3.2.5.1.5.2.2
Use the power rule to combine exponents.
Step 3.2.5.1.5.3
Add and .
Step 3.2.5.1.6
Multiply by .
Step 3.2.5.1.7
Rewrite using the commutative property of multiplication.
Step 3.2.5.1.8
Multiply by by adding the exponents.
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Step 3.2.5.1.8.1
Move .
Step 3.2.5.1.8.2
Multiply by .
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Step 3.2.5.1.8.2.1
Raise to the power of .
Step 3.2.5.1.8.2.2
Use the power rule to combine exponents.
Step 3.2.5.1.8.3
Add and .
Step 3.2.5.1.9
Multiply by .
Step 3.2.5.2
Add and .
Step 3.2.5.3
Add and .
Step 3.3
Since the determinant is non-zero, the inverse exists.
Step 3.4
Set up a matrix where the left half is the original matrix and the right half is its identity matrix.
Step 3.5
Find the reduced row echelon form.
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Step 3.5.1
Multiply each element of by to make the entry at a .
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Step 3.5.1.1
Multiply each element of by to make the entry at a .
Step 3.5.1.2
Simplify .
Step 3.5.2
Perform the row operation to make the entry at a .
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Step 3.5.2.1
Perform the row operation to make the entry at a .
Step 3.5.2.2
Simplify .
Step 3.5.3
Perform the row operation to make the entry at a .
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Step 3.5.3.1
Perform the row operation to make the entry at a .
Step 3.5.3.2
Simplify .
Step 3.5.4
Multiply each element of by to make the entry at a .
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Step 3.5.4.1
Multiply each element of by to make the entry at a .
Step 3.5.4.2
Simplify .
Step 3.5.5
Perform the row operation to make the entry at a .
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Step 3.5.5.1
Perform the row operation to make the entry at a .
Step 3.5.5.2
Simplify .
Step 3.5.6
Multiply each element of by to make the entry at a .
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Step 3.5.6.1
Multiply each element of by to make the entry at a .
Step 3.5.6.2
Simplify .
Step 3.5.7
Perform the row operation to make the entry at a .
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Step 3.5.7.1
Perform the row operation to make the entry at a .
Step 3.5.7.2
Simplify .
Step 3.5.8
Perform the row operation to make the entry at a .
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Step 3.5.8.1
Perform the row operation to make the entry at a .
Step 3.5.8.2
Simplify .
Step 3.5.9
Perform the row operation to make the entry at a .
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Step 3.5.9.1
Perform the row operation to make the entry at a .
Step 3.5.9.2
Simplify .
Step 3.6
The right half of the reduced row echelon form is the inverse.
Step 4
Multiply both sides by the inverse of .
Step 5
Simplify the equation.
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Step 5.1
Multiply .
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Step 5.1.1
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 and the second matrix is .
Step 5.1.2
Multiply each row in the first matrix by each column in the second matrix.
Step 5.1.3
Simplify each element of the matrix by multiplying out all the expressions.
Step 5.2
Multiplying the identity matrix by any matrix is the matrix itself.
Step 5.3
Multiply .
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Step 5.3.1
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 and the second matrix is .
Step 5.3.2
Multiply each row in the first matrix by each column in the second matrix.
Step 5.3.3
Simplify each element of the matrix by multiplying out all the expressions.