Find the following matrices: - 3A + 2B

Question

$A = \begin{bmatrix}3 & 1 & 1 \-1 & 2 & 5\end{bmatrix}, B = \begin{bmatrix}2 & -3 & 6 \-3 & 1 & -4\end{bmatrix}$

Accepted Answer

Hey, everyone. Here we are asked to perform the following matrix operation negative four times A plus five times B. Here we are given the matrices for A and B where both are two by three matrices. The first row for matrix A has the values 254. And the second row has the values negative and four matrix B. The first row has the values one negative 45. And the second row has the values negative 13 and negative five. Here we have four answer choice options, answer choices A through D. Each of them being a different two by three matrix. That is the result of the matrix operation negative four times A plus five times B. So now to begin our matrix operation, we first want to substitute in the two by three matrices for the corresponding A and B variables. So to begin, we have negative four times A. So substituting in matrix A, we have again a two by three matrix with the first row containing the values 254. And the second row has the values negative 310. Then we have plus five times B and the matrix for B has the first row of values one negative four and five. And the second row has the values negative +13 and negative five. So now we want to first multiply A by negative force. That means we take each entry in matrix A and multiply it by negative four. And then we repeat this process for matrix B but instead multiply each entry by five. So now rewriting our matrices, we again multiply each entry of A by negative four. So we have negative four times two as the first entry. Then we have negative four times five, then we have negative four times four. And moving to the second row, we have negative four times negative three, we then have negative four times one. And lastly, we have negative four times zero. Now moving on to the second part of our matrix operation where we now manipulate matrix B, we now have five times each entry in matrix B. So we have five times one, we then have five times negative four, we then have five times five. And now moving to the second row, we have five times negative one. Next we have five times three. Then lastly, we have five times negative five. So now we just want to simplify both matrices. So with our first matrix here multiplying each term by negative four, we get the matrix with the values negative eight, negative 20 negative 16. And the second row is 12 negative four and zero. Now we have plus our second matrix which has now the simplified values of five, negative 20 and 25 the second row is negative 5 15 and 25. So now the next step is to just add our two matrices together. And so that means we need to add each entry in our first matrix with the corresponding entry in the second matrix. So starting in the first row and first column, we have negative A. So then we look to the first row and first column of our second matrix which is five. So in the first row and first column of our adjusted our added matrix, we have negative eight plus five. And so now we move to the second column again, still in the first row and we will have negative negative 20. Then moving to the third column, we have negative 16 plus 25. Now moving down to the second row, looking again at the first column here we have 12 plus negative five. And again, we write this in the now second row and first column of our new matrix. And now moving to the right to the second column, we will have negative four plus 15. And then moving to our last row and last column, we have zero plus 25. So now with all of these expressions here in our new matrix, we just want to simplify by adding all of these values together. And so we are left with the first row having the simplified values of negative three, negative 40 and nine. And the second row takes the values 7-Eleven and 25. So now after performing our matrix operations, we are left with this new simplified matrix leaving us with answer choice C where we see that negative four times A plus five times B gives us the new two by three matrix where the first row has the values negative three, negative 40 and nine and the second row has the values 7-Eleven and 25. Thanks for watching. I hope you found this video helpful and I'll see you again next time.

Find the following matrices: - 3A + 2B
$A = \begin{bmatrix}3 & 1 & 1 \\-1 & 2 & 5\end{bmatrix}, B = \begin{bmatrix}2 & -3 & 6 \\-3 & 1 & -4\end{bmatrix}$

Key Concepts

Matrix Addition and Subtraction

Scalar Multiplication of Matrices

Matrix Dimensions and Compatibility

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