Perform each operation, if possible.

Question

[\begin{matrix} - 2 & 5 & 5 \\ 0 & 1 & 4 \\ 3 & - 4 & - 1 \end{matrix}] [\begin{matrix} 1 & 0 & - 1 \\ - 1 & 0 & 0 \\ 1 & 1 & - 1 \end{matrix}]

Accepted Answer

Hello, everyone. We are asked to apply the given operation on the following matrices. We have 23 by three matrices and we are being asked to multiply them together. Our answer choices are 43 by three matrices. All of which have slightly different values. Recall that to multiply matrices, you must have the columns of the first matrix match the rows of the second matrix. As these are both three by three matrices, we know that does match also from the dimensions of the matrices. We can tell that our answer matrix will also be a three by three as it's from the rows of the first matrix and the columns of the second. So this is gonna get a little long. But I'm gonna try to multiply along the way to multiply matrices. We take the first row of the first matrix and multiply the corresponding elements of the first column of the second matrix and then add them together for the first element of the first row. So we are going to take negative three and multiply it by three as the first element in both the first row of the first matrix and the first column of the second matrix. And then seven as the second element of the first row multiplied by negative three as the second element of the first column. And next seven as the third element of the first row multiplied by three as the third element of the first column. Next, going to the next element in that row, we are going to multiply the first row of the first matrix by the second column of the second matrix. So we'll again have negative three. And now it's gonna be multiplied by zero. We're gonna add to that seven as the second element of the first row multiplied by zero as the second element of the second column. And then we're gonna add seven as the third element of the first row multiplied by three as the third element of the second column of the second matrix. All right. So for our third element of our first row of our answer matrix, we are now multiplying the first row of the first matrix by the third column of the second matrix. So we get negative three, multiplied by negative three plus seven is the second element of the first row multiplied by zero as the second element of the third column plus seven multiplied by negative three, which is the third element of the third column. Going to our next row. We now we're going to do the second row of the first matrix multiplied by each column of the second matrix. So first, we're going to do the second row multiplied by the first column. So the first element of the second row is zero multiplied by the first element of the first column of the second matrix, which is three plus the second element of the second row, which is two multiplied by the second element of the first column of the second matrix which is negative three plus the third element of the second row of the first matrix multiplied by the third element of the first column of the second matrix, which is three next element in that row again will be the second row of the first matrix and now multiplied by the second column of the second matrix. So we get zero multiplied by zero plus for the second element two multiplied by zero. And then adding five multiplied by the third element of that second column which is three, all right. Third element in this row will be the second row of the first matrix multiplied by the third column of the second matrix. So we get zero multiplied by the first element of that third column. So negative three plus two multiplied by the second element of that column, which is zero plus five multiplied by the third element of the third column which is negative three. Once we're done writing all the multiplication, I will do the multiplication out. This seemed like the best way to make sure I have all my numbers where I need them. So next, the final row of our answer matrix will be the third row of the first matrix multiplied by the columns of the second matrix. So starting with the third row of the first matrix. So first elements nine multiplied by the first element of the first column of the second matrix which is three plus negative six is the second element of the third row multiplied by negative three is the second element of the first column of the second matrix plus negative two multiplied by the third element of the first column which is positive three. Next element in that row comes from the third row of the first matrix and the second column of the second matrix. So we have nine multiplied by zero plus negative six multiplied by zero plus negative two multiplied by three. All right, one more element to plan out. And then we can do all the multiplication. So the third row of the first column, um sorry third row of the first matrix multiplied by the third column of the second matrix. So we get nine multiplied by negative three plus negative six multiplied by zero plus negative two, multiplied by negative three. All right, we have all of our values. Now we need to do out the operations. So with my answer matrix down below and so negative three time multiplied by three is negative nine plus a negative 21 plus a positive 21. So this will be negative nine for the first element of the first row going to my second element of the first row of my answer matrix zero times anything is zero. So we only have to really multiply seven and three which is 21. Third element I have negative three multiplied by negative three which is nine uh seven times zero is multiplied by negative three is negative 21. So negative 21 plus a positive nine should give me a negative 12. First row done onto the second row. First element of the second row, zero times anything will cancel out. So two multiplied by negative three is negative six plus five multiplied by three, which is 15. So that would be a positive nine when combined. Next, that middle element zeros will cancel out. So we are left with five multiplied by three which is third element of that row. Again, the zeros will cancel themselves out. So we are left with five multiplied by negative three. So negative 15 final row. Here's where we have some bigger numbers floating around in there. So it might take a little bit more work. First element we have nine multiplied by three which is plus negative six multiplied by negative three, which is a positive 18 and then negative two multiplied by three, which is negative six. Combining all of those numbers, we get 39. Second term in our last row, the zeros will cancel out. The only thing we need to worry about is negative two multiplied by three, which is negative six. Third element, the zero will cancel out. So we have nine multiplied by negative three which is negative 27. And we're gonna add to that negative two multiplied by negative three, which is a positive six. So this should be a negative 21. So our three by three matrix that is our answer has a first row of negative 9 21 negative 12, a second row of 9 15, negative 15 and a third row of 39 negative six, negative 21. And that's answer choice C have a nice day.

Perform each operation, if possible.
$[\begin{matrix} - 2 & 5 & 5 \\ 0 & 1 & 4 \\ 3 & - 4 & - 1 \end{matrix}] [\begin{matrix} 1 & 0 & - 1 \\ - 1 & 0 & 0 \\ 1 & 1 & - 1 \end{matrix}]$

Key Concepts

Matrix Addition and Subtraction

Matrix Multiplication

Dimension Compatibility

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