Find the products AB and BA to determine whether B is the mult...

Question

Find the products AB and BA to determine whether B is the multiplicative inverse of A.

$A = \begin{bmatrix}4 & -3 \-5 & 4\end{bmatrix}, B = \begin{bmatrix}4 & 3 \5 & 4\end{bmatrix}$

Accepted Answer

Hey everyone welcome back in this problem. We are given matrices, A and B. And were asked to work out the product A. B and B. A. In order to show if B is the multiplication inverse of A. Alright, so let's go ahead and start with the dimension. These are both two by two matrices. Ok, So we can multiply them and we know that be could possibly be the multiplication inverse because they have the same dimension. Okay, Alright, so now let's get started with our product A B is going to be one negative two. First column negative 23, 2nd column K for a times B. Which has negative three negative two in its first column negative two negative one in its second column. Now when we're multiplying matrices, what we're gonna do is we're gonna take a row of the first matrix. A column of the second matrix. We're gonna multiply those element by element. Okay, and that's gonna go in the corresponding row and column of our resulting matrix. And what does this look like? Well, let's start with the first row of a. Times the first column of a. Okay, so we have the first element of the row, one times the first element of the column, negative three. Okay then we add and we work our way across the row and down the column. So we get the second element in the row negative two times the second element in the column, negative two. Okay, So this is the result of the red. Now we're in row one, column one. Okay, and so this goes in row one column, one of our resulting matrix. Okay working down the row, we can choose row two and stick to the same column. Column one. Okay, we get the first thing in the row negative two times the first thing in the corresponding column -3. Ad work across the road down the column. The second thing in the row is three times the second thing in the column is -2. All right, now that we've seen these two examples for the first column. Let's go ahead and do this in the next column. Okay, so in the next column we're gonna start with the first row times the second column. Okay, so first row times the second call. We're going to end up with the following the first element of the row, one times the first element of the column, -2. Okay, add second element of the ro negative two times the second element of the column, negative one. Okay, this is coming from those blue rows and columns and finally, if we do the second row in the second column, okay, we're gonna get negative two times negative two. Okay, plus the second element in the row, three times a second element in the column, -1. So now we've done our multiplication, we're down to one matrix. All we need to do now is add and subtract to get our resulting matrix. Alright, so we have one times negative three. So we have negative three plus negative two times negative two is four, so negative three plus four gives us one. Second row. First column we have negative two times negative three. That's gonna be six plus three times negative two negative minus six is going to be zero. Okay, the top right, one times negative two is negative two plus negative two times negative one is two negative two plus two is zero. And in the bottom right, negative two times negative two is four plus three times negative one minus 34 minus three is one. And so our resulting matrix A B is going to be equal to the identity matrix in two dimensions. Alright, so we found A B. Let's go ahead and do the same for B. A. Okay, and I'm gonna leave out the colors because I think hopefully we get the process and we'll see it um without the colors maybe getting in the way. All right, so be a. We're gonna have our B matrix negative three, negative two. In the first column negative two negative one. In the second column A one negative two. In the first column negative 23 in the second column. So row one, column one of our resulting matrix. We're going to take the first row of B times this first column of A. Okay, so the first element in the row is negative three times the first element in the column one. We add the next element in the row negative two times the next element in the column -2. Moving down to row two column one. First element in the row negative two. First element in the column one multiply them and then add the second element in the row negative one times the second element in the column negative two. Okay, moving across to column two, starting with row one. The first element in the row is negative three times the first element in the column negative two plus first element or sorry, second element in the road negative two times second element in the column three. And finally second row. Second column first element in the row negative two times first element in the column negative two plus. Second element in the row negative one times second element of the column three. Alright, so again, we've done the multiplication. The hard part is over. We just need to add subtract and do some simple arithmetic here. So we have negative three plus four. That's gonna give us one negative two plus 20. Six minus six is going to give us zero. Okay, First row, second column and second row. Second column we have four minus three. That's gonna give us one. And what you'll see is that here, B. A. We also have the identity matrix and two dimensions. Okay. All right. So now if we're going to answer that question is B. The multiplication inverse of A. We'll notice that the product A. B. That we found is equal to the product B. A. That we found, and they are both equal to the identity matrix In two dimensions. Okay. This means that B is the multiplication inverse of a. Okay, just use short form there. Alright. So if we go back up answer this question, we found that A B is equal to B. A. They're both equal to the sorry to the identity matrix and two dimensions. And so the answer is A B is the multiplication inverse of a. Thanks everyone for watching. I hope this video helped see you in the next one.

Find the products AB and BA to determine whether B is the multiplicative inverse of A.
$A = \(\begin{bmatrix}\)4 & -3 \\-5 & 4\(\end{bmatrix}\), B = \(\begin{bmatrix}\)4 & 3 \\5 & 4\(\end{bmatrix}\)$

Key Concepts

Matrix Multiplication

Multiplicative Inverse of a Matrix

Identity Matrix

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