In Exercises 37 - 44, perform the indicated matrix operations ...

Question

In Exercises 37 - 44, perform the indicated matrix operations given that A, B and C are defined as follows. If an operation is not defined, state the reason.

$A = [\begin{matrix} 4 & 0 \\ - 3 & 5 \\ 0 & 1 \end{matrix}], B = [\begin{matrix} 5 & 1 \\ - 2 & - 2 \end{matrix}], C = [\begin{matrix} 1 & - 1 \\ - 1 & 1 \end{matrix}]$

A(BC)

Accepted Answer

consider the following agencies that perform the indicated operation. We have three matrices here and the operation is C times a times B. So we can multiply these as long as the number of columns in the first matrix are equal to the number of rows and the second matrix, in our case We have to buy two for a. And the two x 2 for B. So they both have the same number of rows and columns. So we can multiply those together. Let's go ahead and find a. b. 1st. Now with the matrix we're first gonna multiply our top row in a by our first column in B stone. Our top round a is 3, 4. Well three multiplied by the first part of B, which is two. Now we add our next part which is four multiplied by our column. Our 2nd row in that calm is eight. So that gives us that was our 34 and our 28. Now our next one. We would take sorry we would take the top rope of our first matrix A multiply it by the second column of our matrix B. Just 4 -1. So for the next part you would have three times And then plus four times negative one. Now for the bottom two We have our bottom row of a Native two and 1 multiplied by our second column or my first column of B. 28 Which would give us -2 times two Plus one times 8. Finally we would have our second row negative 21 multiplied by our second column of B or negative one, Which we would get there two times four And Plus one times negative one. And then we can simplify. So we get three times 26 plus 32. It's 38 12 - is eight negative four plus And -8 -1, -9. So A. B. Is this matrix now multiply it by matrix C. We noticed here, we said that we have to have the same number of rows in the center of columns and C. And A. B. And by that, I mean we're multiplying by C. First we have to have two columns and A B has to have two rows which it does. So C. Times A B. We can do that. And so now you want to look at 5679, 3, 0 Multiplied with 8 or -9. So much like what we just did or multiply our top row by our first column. So you'll make matrix, I'll make slow wider. Our first one we have five times Plus six times 4. Our top road times our first column. Next one We have our top row and the second column five times eight Plus six times -9. Now we have our miller rho times our first column seven times Plus nine times 4. And our miller roads under second column seven times eight Plus nine times -9. And then we have our last call, our last row times our first column which is three times Plus zero times 4 and three times 8 Plus zero Times 9. Now we can go ahead and simplify Top one. Let's say we get five times 38 1 90 plus 40 minus 2 66 plus 56 minus 1, 14, 0 And 24 plus zero. Finally, we can simplify those. He would get 2 14 negative 14 name 25 1 14 and 24. Yeah. Simplify any further, which means the mandatory matrix. Is this the answer to our problem? It's answer. B. Okay. I hope to help you solve the problem. Thank you for watching. Goodbye.

Key Concepts

Matrix Multiplication

Matrix Dimensions and Compatibility

Associative Property of Matrix Multiplication

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