If possible, find AB, BA, and A^2. (If not possible, enter...

If possible, find AB, BA, and $A^2$. (If not possible, enter IMPOSSIBLE.) $\begin{bmatrix} 2 & -2 \ -4 & 0 \ 9 & 6 \end{bmatrix}$, $B = \begin{bmatrix} 1 & 0 \ 0 & 1 \end{bmatrix}$ (a) AB $AB = \begin{bmatrix} \boxed{} & \boxed{} \\ \boxed{} & \boxed{} \end{bmatrix}$ ?? (b) BA $BA = \begin{bmatrix} \boxed{} & \boxed{} \\ \boxed{} & \boxed{} \end{bmatrix}$ ?? (c) $A^2$ $A^2 = \begin{bmatrix} \boxed{} & \boxed{} \\ \boxed{} & \boxed{} \end{bmatrix}$

Transcript

00:01 Given the matrices a equal 1 -4 -5 -4 and b equal to negative 1, negative 1 -7, we want to find or calculate a times b, b times a and a -square, if it's possible.

00:21 So in this case, a and b both are matrices of real numbers of order 2 by 2.

00:31 That is, we have two rows and two columns in each of these matrices.

00:38 For that reason, we can calculate all the products we have here.

00:45 That is, we can calculate ab, b, a, and a square, which is by definition a times a.

00:54 That is because being the matrix 2 by 2 and the other matrix 2 by 2, the condition we need to multiply the two matrices.

01:07 That is the number of columns of the first factor equal the number of rows of the second factor will always be true for a times b, b times a and a square.

01:20 So we can calculate all these products here.

01:29 So let's do first can be calculated.

01:41 First, let's calculate a times b, that is 1 ,45, 4 times 2, negative 1, negative 1, 7.

01:57 And we know this is done this way.

02:01 First row of the first factor, that is the first row of a, is multiplied by the first column of b like an internal or dot product.

02:17 That is we do 1 times 2 plus 4 times negative 1.

02:29 That's the first entry of the first row of the product.

02:35 Then we do 1 times negative 1 plus 4 times 7.

02:47 And that will be the second element in the first row.

02:52 Now we go to the second row.

02:54 Would be the second row of a times each of the rows of b.

02:59 So first element, the second row will be 5 times 2 plus 4 times negative 1.

03:12 And the second element in the second row of the product will be 5 times negative 1 plus 4 times 7.

03:24 And this is the resulting matrix, which in this case is equal to the first element in the first row will be 2 minus 4.

03:36 The second element in the second row will be negative 1 plus 28.

03:42 Then the second row will be 10 minus 4 and the last element will be negative 5 plus 28...

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