Are the given matrices inverses of each other? (Hint: Check to...

Question

Are the given matrices inverses of each other? (Hint: Check to see whether their products are the identity matrix I_n.)

$[\begin{matrix} 5 & 7 \\ 2 & 3 \end{matrix}]$

Accepted Answer

Hello. Today we're going to be determining whether the two given matrices are inverses of each other. Let's start by labeling the first matrix as matrix A and we'll be labeling the se second matrix as matrix B. In order to determine whether matrix A or matrix B are inverses of each other. We're going to need to take the product of both of the matrices. If the product of matrix A and matrix B result in the identity matrix for A two by two matrix C, then that will prove that they are inverses of each other. Now, the identity matrix for a two by two matrix is given to us as the elements 10 in its first row and 01 in its second row. So if the product gives us this exact matrix that will prove that the two matrices are inverses of each other. Let's start by setting up the multiplication. We're going to write matrix A which is given to us as 11 and six in its first row and nine and five in its second row. And we'll be multiplying this by matrix B which is given to us as five and negative six in its first row with negative nine and positive 11 in its second row. Now the question is how do we multiply these two matrices together? Well, we're going to need to solve for the independent elements in the new product. So if we want to find the first element which is located in the first row, we're going to need to multiply every element in the first row of A with every element in the first column of B and we'll be taking the sum of those products. So to start with the first element, which is in the first row, first column, we're going to multiply 11 with 511, multiplied by five will give us the value of 55 and we'll be adding the product of six and negative nine, which will give us negative 54. Thus, this will be the quantity that expresses the first element in the first row and first column of the new matrix. Now, if we want to find the second element in the first row, we'll be multiplying the first row of A with the second column of B and taking the sum of those products. So 11 multiplied by negative six will give us the value of negative 66 and six multiplied by 11 will give us the value of positive 66. Now we're going to move on to the second row in order to identify the first element in the second row, we're going to multiply the second row of A with the first column of B and take the sum of those products. So nine multiplied by five will give us the value of 45 and five multiplied by negative nine will give us the value of negative 45. And to identify the last element in the second row, we're going to multiply the second row of A with the second column of B. So nine multiplied by negative six will give us the value of negative 54 and five multiplied by positive 11 will give us the value of positive 55. Now, what we need to do is we need to simplify each of the elements in the new matrix. 55 minus 54 will give us the value of positive one negative 66 plus 66 will give us the value of zero 45 minus 45 will give us the value of zero and negative 54 plus 55 will give us the value of positive one. So this is going to be the product of matrix A and matrix B. And as you can see the product of the two matrices match up with the identity matrix for a two by two matrix. And with that being said that proves that matrix A and matrix B are inverses of each other. And the answer to this problem is going to be a. So I hope this video helps you in understanding how to approach this problem. And I'll go ahead and see you all in the next video.

Are the given matrices inverses of each other? (Hint: Check to see whether their products are the identity matrix I_n.)
$[\begin{matrix} 5 & 7 \\ 2 & 3 \end{matrix}]$

Key Concepts

Matrix Multiplication

Identity Matrix

Matrix Inverse

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