13.6 MATRIX SOLUTION OF A LINEAR SYSTEM

Examine the matrix equation below.

  How would you solve for X?

In order to solve this type of equation, we need the of the matrix that is multiplied by X.

A matrix times it’s inverse is the matrix:

5 27 3

⎡

⎣⎢

⎤

⎦⎥X = 9 −6

−3 2

⎡

⎣⎢

⎤

⎦⎥

inverse

identity

1 00 1

⎡

⎣⎢

⎤

⎦⎥

Inverse Matrix

The inverse of 2X2 matrix is found by the following:

If the determinant of A is , then there is no inverse and we call this a singular matrix.

A =a1 b1

a2 b2

⎡

⎣

⎢⎢

⎤

⎦

⎥⎥

A−1 =1

detA

b2 −b1

−a2 a1

⎡

⎣

⎢⎢

⎤

⎦

⎥⎥

zero

Find the inverse of the matrix. If the matrix is singular, state so.

1. 2.

2 34 7

⎡

⎣⎢

⎤

⎦⎥

−11 5−4 2

⎡

⎣⎢

⎤

⎦⎥

Solving Matrix Equations

We use inverses when solving matrix equations like the opening example.

For matrix equation , in order to get X by itself we need to LEFT multiply both sides of the equation by the inverse of A ( ).

This would look like .

AX =B

A−1

A−1AX =A−1B

3. Solve the opening example

5 27 3

⎡

⎣⎢

⎤

⎦⎥X = 9 −6

−3 2

⎡

⎣⎢

⎤

⎦⎥

Solve for matrix X.

4.

6 −35 −2

⎡

⎣⎢

⎤

⎦⎥X = 9 −6

12 −3

⎡

⎣⎢

⎤

⎦⎥

Solving a System

We can also use this method for solving a system of equations.

For the following system,

  The system in matrix form would look

like:

−4x+ y=−11

−5x+ 3y=−19

−4 1−5 3

⎡

⎣⎢

⎤

⎦⎥

xy

⎡

⎣⎢⎢

⎤

⎦⎥⎥= −11

−19

⎡

⎣⎢

⎤

⎦⎥

In order to solve the system we need to multiply both sides by the of the coefficient matrix. Solve the system!

inverse

−4 1−5 3

⎡

⎣⎢

⎤

⎦⎥

xy

⎡

⎣⎢⎢

⎤

⎦⎥⎥= −11

−19

⎡

⎣⎢

⎤

⎦⎥

5. Solve the system using a matrix equation.

3x + 2y=−3

7x + 4y=1