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Solving Matrix Equations

A matrix equation is an equation in which a variable stands for a matrix .

You can solve the simpler matrix equations using matrix addition and scalar multiplication .

Examples 1:

Solve for the matrix X : X + [ 3 2 1 0 ] = [ 6 3 7 1 ]

X + [ 3 2 1 0 ] [ 3 2 1 0 ] = [ 6 3 7 1 ] [ 3 2 1 0 ] X + [ 0 0 0 0 ] = [ 6 3 3 2 7 1 1 0 ] X = [ 3 1 6 1 ]

Examples 2:

Solve for the matrix X : X [ 9 3 6 0 ] = [ 4 0 12 10 ]

X [ 9 3 6 0 ] = [ 4 0 12 10 ] X [ 9 3 6 0 ] + [ 9 3 6 0 ] = [ 4 0 12 10 ] + [ 9 3 6 0 ] X [ 0 0 0 0 ] = [ 4 + ( 9 ) 0 + ( 3 ) 12 + 6 10 + 0 ] X = [ 5 3 18 10 ]

Solving systems of linear equations using matrices:

Matrix equations can be used to solve systems of linear equations by using the left and right sides of the equations.

Examples 3:

Solve the system of equations using matrices:  { 7 x + 5 y = 3 3 x 2 y = 22

  7 x + 5 y = 3 3 x 2 y = 22 [ 7 x + 5 y 3 x 2 y ] = [ 3 22 ]

Write the matrix on the left as the product of coefficients and variables.

[ 7 5 3 2 ] [ x y ] = [ 3 22 ]

coefficient variable constant matrix matrix matrix

First, find the inverse of the coefficient matrix.  The inverse of [ 7 5 3 2 ] is

1 7 ( 2 ) ( 3 ) ( 5 ) [ 2 5 3 7 ] = 1 29 [ 2 5 3 7 ] = [ 2 29 5 29 3 29 7 29 ]

Next, multiply each side of the matrix equation by the inverse matrix .  Since matrix multiplication is not commutative, the inverse matrix should be at the left on each side of the matrix equation.

[ 2 29 5 29 3 29 7 29 ] [ 7 5 3 2 ] [ x y ] = [ 2 29 5 29 3 29 7 29 ] [ 3 22 ]

[ 1 0 0 1 ] [ x y ] = [ 4 5 ]

The identity matrix on the left verifies that the inverse matrix was calculated correctly.

[ x y ] = [ 4 5 ]

The solution is ( 4 , 5 ) .