Section 8.3 the Inverse of a Square Matrix

Section 8.3 • The Inverse of a Square Matrix 149 Name______Section 8.3 The Inverse of a Square Matrix

Objective: In this lesson you learned how to find the inverses of matrices and use inverse matrices to solve systems of linear equations.

Important Vocabulary Define each term or concept.

Inverse of a matrix Let A be an n × n matrix and let In be the n × n identity matrix. If –1 –1 –1 –1 there exists a matrix A such that AA = In = A A then A is called the inverse of A.

I. The Inverse of a Matrix (Page 602) What you should learn How to verify that two To verify that a matrix B is the inverse of the matrix A, . . . matrices are inverses of show that AB = I = BA. each other

II. Finding Inverse Matrices (Pages 603−605) What you should learn

How to use Gauss-Jordan If a matrix A has an inverse, A is called invertible or elimination to find the nonsingular. Otherwise, A is called singular . inverses of matrices

To have an inverse, a matrix must be square . Not all square matrices have inverses. However, if a matrix does have an inverse, that inverse is unique .

To find the inverse of a square matrix A of order n, . . . write the n × 2n matrix that consists of the given matrix A on the left and the n × n identity matrix I on the right to obtain [A : I]. Then, if possible, row reduce A to I using elementary row operations on the entire matrix [A : I]. The result will be the matrix [I : A−1]. If this is not possible, then A is not invertible. Finally, check your work by multiplying to see that AA−1 = I = A−1A. ⎡1 2 4⎤ Example 1: Find the inverse of the matrix A = ⎢1 0 2⎥ . ⎢ ⎥ ⎣2 3 6⎦

⎡ − 3 0 2 ⎤ −1 A = ⎢ − 1 − 1 1 ⎮ ⎣ 3/2 1/2 − 1⎦

III. The Inverse of a 2 × 2 Matrix (Page 606) What you should learn How to use a formula to ⎡a b⎤ If A is a 2 × 2 matrix given by A = , then A is invertible if find the inverses of 2 × 2 ⎣⎢c d⎦⎥ matrices and only if ad − bc ≠ 0 . Moreover, if this condition is true, the inverse of A is given by: ⎡ ⎤ −1 1 d − b A = ⎢ ⎥ ⎢ ⎥ ad − bc ⎣− c a ⎦

The denominator is called the determinant of the 2 × 2 matrix A.

⎡ 3 9⎤ Example 2: Find the inverse of the matrix B = . ⎣⎢− 2 − 7⎦⎥

−1 B = ⎡ 7/3 3 ⎤ ⎣ − 2/3 − 1⎦

IV. Systems of Linear Equations (Page 607) What you should learn How to use inverse If A is an invertible matrix, the system of linear equations matrices to solve systems represented by AX = B has a unique solution given by of linear equations X = A−1B .

Example 3: Use an inverse matrix to solve (if possible) the system of linear equations: ⎧12x + 8y = 416 ⎨ ⎩3x + 5y =152

The solution is x = 24 and y = 16.

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