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.)

$[\begin{matrix} 0 & 1 & 0 \\ 0 & 0 & - 2 \\ 1 & - 1 & 0 \end{matrix}]$

Accepted Answer

Hello. Today we're going to be determining whether the two given matrices are inverses of each other. We're going to label the first matrice C as matrix C A and we'll label the second matrix C as matrix CB. Now, in order to determine whether these two matrices are inverses of each other, if we take the product of matrix C A with matrix CB, then the resulting product should give us the identity matrix for a three by three matrix. The identity matrix 43 by three matrix has the first row of elements as 100. The second row of elements as 010 and the third row of elements as 001. Now again, if the product of A and B gives us this exact matrix that proves that the matrices are inverses of each other. So in order to determine this, let's first start by setting up the multiplication. The first row in matrix A is given to us as 10, negative three. The second row is negative 120 and the third row is 10 negative one. And we're going to be multiplying matrix B which has the first row as 010, the second row as 011 and the third row as negative three, negative two and one. Now the question is how do we multiply 23 by three matrices with each other? Well, we're going to need to identify or figure out what the value of each element in the resulting matrix C is going to be. So we're going to start by finding the element in the first row, first column. Now, in order to find the element in the first row, first column, we're going to multiply every number in the first row of matrix C A with every number in the first column of matrix CB. And we'll be taking the sum of those products. So if we want to identify the first element in the new matrix, we're going to start by multiplying one in the first row of A with zero in the first column of B, one multiplied by zero will give us the value of zero. Then we'll be adding the next product which will be the second element in the first row of A which is zero multiplied by the second number in the first column of B which is zero zero, multiplied by zero will give us the value of zero. Then we'll be multiplying the third number in the first row of A which is negative three with the third number in the first column of B which is negative three, negative three multiplied by negative three will give us the value of positive nine. So we'll be adding nine to the other products. Then if we want to identify the second number in the first row of the new matrix, we're going to multiply the first row of matrix A with the second column of matrix B. And we're going to repeat the process of multiplying each respective element and taking the sum of those products. So one multiplied by one will give us the value of one, then zero, multiplied by one will give us the value of zero and negative three multiplied by negative two will give us the value of positive six. That will be the second number in the first row of the new matrix. Now for the third number in the first row of the new matrix, we're going to multiply the first row of A with the third column of B. So one multiplied by zero will give us the value of zero zero, multiplied by one will give us the value of zero and negative three multiplied by one will give us the value of negative three. Those are going to be the sums that give us the first row of the new matrix. Now we're going to move on to the second row of the new matrix. Now, in order to get the first number in the second row, we're going to multiply the second row of matrix A with the first column of matrix B so negative one multiplied by zero will give us the value of zero two, multiplied by zero will give us the value of zero and zero multiplied by negative three will give us the value of zero. Then we're going to move on to the second number. In the second row of the new matrix, we'll be multiplying the second row of A with the second column of B. So negative one multiplied by one will give us the value of negative one, two multiplied by one will give us the value of two and zero multiplied by negative two. I'm sorry, zero in the second row of a multiplied by negative two will give us the value of zero. Then we move on to the third number in the second row and we'll multiply the second row of A with the third row through a column of B. So negative one multiplied by zero will give a zero two multiplied by one, will give us the value of two and zero multiplied by one will give us the value of zero. Then we're going to move on to the final row of the new matrix. To get the first element, we'll be multiplying the numbers in the third row of A with the numbers in the first column of B. So one multiplied by zero will give us the value of zero zero, multiplied by zero will give us the value of zero and negative one multiplied by negative three will give us the value of positive three. Next, we're going to multiply the third row of A with the second column of B. So one multiplied by one will give us the value of one zero, multiplied by one will give us the value of zero and negative one multiplied by negative two will give us the value of positive two. Finally, for the last element, we're going to be multiplying the third row of A with the third column of B. So one multiplied by zero will give us the value of zero zero, multiply by one will give a zero and negative one multiplied by positive one will give us the value of negative one. Now, in order to simplify the remaining product, we're going to need to simplify each of the elements. So zero plus zero plus nine will give us the value of nine in the first row, one plus zero plus six will give us the value of seven for the second number in the first row, zero plus zero minus three will give us the value of negative three. In the first row zero plus zero plus zero will give us the value of zero in the second row, negative one plus two plus zero will give us the value of positive one. In the second row zero plus two plus zero will give us the value of positive two in the second row zero plus zero plus three will give us the value of three in the third row, one plus zero plus two will give us the value of three in the third row and zero plus zero minus one will give us the value of negative one in the third row. This is going to be the resulting matrix by multiplying matrix A and matrix B together. But as you can see, this matrix does not match the identity matrix. Therefore, this shows that matrix E and matrix A and matrix B are not inverses of each other. And with that being said, the answer to this problem is going to be B. 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.)
$[\begin{matrix} 0 & 1 & 0 \\ 0 & 0 & - 2 \\ 1 & - 1 & 0 \end{matrix}]$

Key Concepts

Matrix Multiplication

Identity Matrix

Matrix Inverse

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