Perform each operation, if possible.

Question

[\begin{matrix} 3 & 2 & - 1 \\ 4 & 0 & 6 \end{matrix}] [\begin{matrix} - 2 & 0 \\ 0 & 2 \\ 3 & 1 \end{matrix}]

Accepted Answer

Hello, everyone. We are asked to apply the given operations on the following matrices. We are given a two by three matrix with a top row of 81 negative three in the second row of 607. And that will be multiplied by a three by two matrix with a first row of negative 40, a second row of 04 and a third row of 75. We're given four answer choices all of which are two by two matrices with slightly varying values. First recall that in order to multiply matrices, we need to check that the columns of the first matrix match the rows of the second matrix. So we have a two by three matrix and a three by two matrix. So the inner numbers do match the columns from the first matrix match the rows of the second. So we can multiply this. Also this information will tell us that our answer matrix will also be a two by two. We can tell by the rows of the first matrix and the columns of the second. So starting on my two by two matrix, I'm gonna write all the multiplication out first and do that in a separate step. So for the first element of my new matrix, I need to multiply the first row of the first matrix by the first column of the second matrix. So I'm gonna match my elements. So the first element of the first row is eight multiplying it by the first element of the first column of the second matrix which is negative four adding to that the second element of the first row, which is one multiplied by the second element of the first column which is zero. And then we're gonna add negative three as the third element of the first row of the first matrix multiplied by seven as the third element of the first column of the second matrix, I'm gonna continue horizontally. So my next element will come from the first row of the first matrix and the second column of the second matrix. So we all have eight as the first element of the row multiplied by zero as the first element of the column added to one as the second element of the row multiplied by four as the second element of the second column. And we're gonna add negative three as the third element of the first row multiplied by five as the third element of the second column moving down to my second row. My multiplication for this element will be the second row of the first matrix and the first column of the second matrix So the first element of the second row is six, we're gonna multiply that by negative four as the first element of the first column of the second matrix. And then we have zero as the second element of that second row multiplied by zero as the second element of the first column of the second matrix. And we're gonna add to that seven as the third element of the second row multiplied by seven as the third element of the first column of the second matrix. All right, one more element to find. And that's gonna be from the second row of the first matrix multiplied by the second column of the second matrix. So we have six multiplied by zero plus our second elements. So zero multiplied by four plus the third elements of seven multiplied by five. All right, we have all of our multiplication written out. Now we are going to multiply and add and get our final answer. So starting in the top left corner, eight multiplied by negative four is negative negative 32 will be added to let's see the zero cancels negative 21. So that will give us a negative 53. Next element over eight multiplied by zero will disappear. Four, multiplied by one is four, adding to that negative three multiplied by five, which is negative 15. So negative 15 plus four is negative going down to my second row. I have six multiplied by negative four. So negative 24 the zeros are zero. And I'm gonna add to that seven multiplied by seven, which is 49. And when I add those together, I get 25. And last element six multiplied by zero is 04, multiplied by zero is zero. So seven multiplied by five is 35. So this is our two by two answer matrix. The top row is negative 53 negative 11. The second row is 25 35 that matches answer choice. D have a nice day.

Perform each operation, if possible.
$[\begin{matrix} 3 & 2 & - 1 \\ 4 & 0 & 6 \end{matrix}] [\begin{matrix} - 2 & 0 \\ 0 & 2 \\ 3 & 1 \end{matrix}]$

Key Concepts

Matrix Dimensions and Compatibility

Matrix Multiplication Process

Resulting Matrix Dimensions

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