Let

Question

Let

A = [\begin{matrix} - 3 & - 7 \\ 2 & - 9 \\ 5 & 0 \end{matrix}]

and

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

. Solve each matrix equation for X. B - X = 4A

Accepted Answer

Hey everyone in this problem. We are asked to consider the following matrices K. Were given a matrix A and matrix B. We're asked to solve the matrix equation for x. Okay. The matrix equation we're given is to be minus X is equal to five A. Alright, so starting off we have to be minus X is equal to five. They were asked to solve for X. So let's go ahead right off the bat and isolate for X. Okay. It's a lot easier to do it when we have this equation like this, then once we write out all of the matrices OK. It's easier just to move it around now and isolate for X. So if we do that, we're going to find the X is equal to two B minus five A. Alright, so substituting in our matrices, this is going to give us two terms of matrix B which has first column negative 975 and second column negative 200. Okay. And we're gonna subtract five times the matrix A. Which has first column negative 610. And second column negative four negative eight and three. Alright, so we have subtraction and we have some multiplying by a constant. Okay, when you have an equation like this, you want to do the multiplication first. Just like you would in a normal equation. Okay, so let's go ahead and do the multiplying of those constants first, when we multiply a matrix by a constant, we can bring the constant inside of the matrix and it's going to multiply to every single element of that matrix. Okay. And so with two times b we're gonna end up with two times -9. two times 7. two times 5 in the first column. Two times negative 22 times 02 times zero in the second column. Okay. And similarly for five times eight, Okay, we're gonna end up with five times negative times 15 times zero in the first column, and five times negative 45 times negative eight. And five times three in the second column. Alright, so now we brought those constants inside. We're just left with two matrices being subtracted. Let's just go ahead and simplify within each of those matrices. Before we do the subtraction. Okay, so the first matrix we're going to have negative 18, 14 and 10. In the first column negative 400 In the second column. Okay. And in the second matrix. First column we're gonna have negative 0. And in the second column negative 20 negative and 15. Okay. Alright, now we have two matrices and we need to subtract them. Okay, when you're subtracting matrices, you want to double check that The dimension is the same? Okay. It needs to be the same. In this case. We have 23 by two matrices. So the dimension is the same. So we can do the subtraction. We're doing subtraction of matrices. This is going to be an element wise operation. What that means is we're gonna take an element in the first matrix we're going to subtract the corresponding element in the second matrix, Okay, so let's see this in action. Okay, So we're gonna take first row first column The first matrix we have negative 18. And the second matrix we have negative 30. Okay, we're going to subtract the two. So we get negative 18 minus negative 30. We're gonna put it in the exact same position in the resulting matrix. So it's gonna go row one column one. Okay, let's move down in the column. So in row two column one we have 14 in the first matrix five. In the second matrix. So in row two column one of our resulting matrix we're gonna have minus five. All right, we're going to continue with the same thing for each of the other terms. So we're gonna have 10 0. Okay? In the second column negative four minus negative 20. Zero minus negative 40 & 0 -15. And this is going to give us a resulting matrix simplifying. Within our matrix negative 18 minus negative 30. That's going to be 12. 14 minus five gives us 9, 10 minus zero, gives us 10 negative four minus negative 20 is going to give us 16 0 minus negative 40 gives us 40 and zero minus 15 gives us negative 15. Okay, And so this is the value of X that we were looking for. Okay, it's this three by two. Matrix with first row 12 16. 2nd row nine 43rd row 10 negative 15. And if we scroll back up to our answer choices, that is going to correspond with answer B. Thanks everyone for watching. I hope this video helped see you in the next one.

Let $A = [\begin{matrix} - 3 & - 7 \\ 2 & - 9 \\ 5 & 0 \end{matrix}]$ and $B = [\begin{matrix} - 5 & - 1 \\ 0 & 0 \\ 3 & - 4 \end{matrix}]$ . Solve each matrix equation for X. B - X = 4A

Key Concepts

Matrix Addition and Subtraction

Scalar Multiplication of Matrices

Solving Matrix Equations

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