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**Finding the Inverse of a Matrix**

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**Properties of Matrices**

We have discovered that the commutative property for multiplication does not work for matrix multiplication. Let’s consider some of the other properties of real numbers. Is there a multiplicative identity for matrices? Is there a multiplicative inverse for matrices?

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**The Multiplicative Identity**

The multiplicative identity for real numbers is the number 1. The property is: If a is a real number, then a x 1 = 1 x a = a. In terms of matrices we need a matrix that can be multiplied by a matrix (A) and give a product which is the same matrix (A).

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**The Multiplicative Identity**

This matrix exists and it is called the identity matrix. It is named I and it comes in different sizes. It is a square matrix with all 1’s on the main diagonal and all other entries are 0.

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**The Multiplicative Identity**

Multiply AI a11= (-2)(1) + (5)(0) = -2 a12= (-2)(0) + (5)(1) = 5 a21= (4)(1) + (0)(0) = 4 a22= (4)(0) + (0)(1) = 0

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**The Identity Matrix for Multiplication**

Let A be a square matrix with n rows and n columns. Let I be a matrix with the same dimensions and with 1’s on the main diagonal and 0’s elsewhere. Then AI = IA = A

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**The Multiplicative Identity**

Give the multiplicative identity for matrix B. This identity matrix is I4.

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**The Multiplicative Inverse**

For every nonzero real number a, there is a real number 1/a such that a(1/a) = 1. In terms of matrices, the product of a square matrix and its inverse is I.

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The Inverse of a Matrix Let A be a square matrix with n rows and n columns. If there is an n x n matrix B such that AB = I and BA = I, then A and B are inverses of one another. The inverse of matrix A is denoted by A-1.

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The Inverse of a Matrix To show that matrices are inverses of one another, show that the multiplication of the matrices is commutative and results in the identity matrix. Show that A and B are inverses.

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The Inverse of a Matrix and

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The Inverse of a Matrix

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**Finding the Inverse of a Matrix - Method 1**

Use the equation AB = I. Write and solve the equation:

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Inverses – Method 1, cont.

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**Inverses – Method 1, cont. So the inverse of A =**

We can check this by multiplying A x A-1

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**Finding the Inverse with a Calculator**

To find the inverse of a matrix using the calculator, enter the matrix into the calculator and use the x-1 key.

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**Finding the Inverse with a Calculator**

Find the inverse of each matrix using the calculator.

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**Finding the Inverse with a Calculator**

This error message means that the matrix does not have an inverse. A matrix that does not have an inverse is called an invertible matrix.

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Determinants Each square matrix can be assigned a real number called the determinant of the matrix. It is denoted by the symbol means the determinant of A.

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Determinants The determinant of a 2 x 2 matrix is found as follows:

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Determinants Find the determinant of the matrix.

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**Determinants Find the determinant of the matrix.**

If the determinant of a matrix = 0, the matrix does not have an inverse. Matrix H is invertible.

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**Determinants can be used to find the inverse of a matrix.**

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**Determinants can be used to find the inverse of a matrix.**

is called the adjoint of the original matrix. Notice it is found by switching the entries on the main diagonal and changing the signs of the entries on the other diagonal.

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**Find the multiplicative inverse of:**

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**We can check to see if we are correct by multiplying**

We can check to see if we are correct by multiplying. Remember that AA-1 = I

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**Find the inverse using determinants.**

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**Find the inverse No inverse**

Recall that when the determinant of a matrix is 0 the matrix will not have an inverse because division by 0 is undefined.

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**Finding the determinant of a 3 x 3 matrix**

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**Finding the determinant of a 3x3 matrix.**

One way to find the determinant of a 3x3 matrix is the formula below.

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**Find the determinant using the formula**

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**Find the determinant using the formula**

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**Find the determinant using the formula**

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**Find the determinant using the formula**

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