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. 3X + 2A = B

Accepted Answer

Hey everyone in this problem. We are asked to solve the following matrix equation for X. And were given the matrices A. And B. So we have two X plus four. A. Is equal to B. two x plus four a. Is equal to be okay. And we want to solve for X. So what we're gonna do is we're going to isolate X right off the bat here before we start working with our matrices. And that's just gonna make it simpler later. Okay, We won't have to rearrange all of our matrices. We'll do the rearranging now. So in order to isolate X, we get to X. We move the negative or sorry, the foray to the right hand side. We get B minus four A. Okay. And dividing by two, we're going to get one half B minus two. A. Alright, great. So we've isolated for X now. Let's go ahead and substitute in our matrices. So we're gonna have X. Is equal to one half times the matrix B. Which is a three by two matrix with first calm negative seven, sorry, negative 975. And second column negative 200. Okay minus two times matrix A. Which again is a three by two matrix with first column negative six, one zero and second column negative four negative 83. Alright, so we have some constant multiplication of the matrices and we have subtraction. Okay, what we wanna do is we want to deal with the multiplication first. Just like we would in a normal equation. Okay, so when we're multiplying a matrix by a constant, we're gonna take that constant inside of the matrix. And it's gonna multiply every single element. Okay, So, what we're gonna have here Is we're going to have negative 9/2. 7/2. 5/2. Negative 2/2. 0/2. 0/2. Okay, for our first matrix and our second matrix, we're going to do the same thing. That constant two is going to come inside and multiply every single element. So, we have two times negative 62 times 12 times zero. two times negative four, Two times negative eight and 2 times three. All right. So now we don't have any terms multiplying on the outside. We just have two matrices being subtracted. Let's just simplify within those matrices. Before we do the subtraction, I'm gonna give ourselves some more space here to work. Alright, so, the first matrix negative 9 to 7/2. 5/2. In the first column. Okay, negative two divided by negative two. That's just going to be very negative. Two divided by two is gonna be negative 100 in the second column. In the second matrix we're gonna get negative 12 to 0 In the first column negative eight negative 16. And six in the second call. All right. So, we've got our first matrix we've got our second matrix. Now we can subtract the two. When we're subtracting we want to double check that the matrices have the same dimension. Okay, they must have the same dimension to do the subtraction. In this case they're both three by two matrices. So we're okay to do the subtraction. We're going to subtract element wise. What that means is we're going to take an element from a we're going to subtract the corresponding element from the second matrix. I'm sorry when I said an element from matrix A. I meant from the first matrix. All right so let's see this in practice. So again take the first row. First column okay we get negative nine over to from the first matrix first row first column of the second matrix is negative 12. So in the first row and first column of our resulting matrix we're going to get negative 9/2 minus negative 12. We're gonna do the same for the next one. So second row first column we get 7/2. In the first matrix we have to in the second matrix. So the second row first column of our resulting matrix is going to have 7/2 minus two. And we can do the same for the rest of the terms. So third row first column we're gonna get 5/2 minus zero. And then in the second column we're gonna get negative one minus negative 80 minus negative 16. N 0 -6. And this is gonna give us a matrix. Remember this is x. This is gonna give us a matrix negative 9/2 -20. Hey this is gonna give us 15 over to 7/2 minus two. Okay so 7/2 minus 4/2. This is going to give us 3/2 And in the bottom we get 5/2. Okay? In the second column we're gonna get 7 16 and negative six. And so the value of X. From the equation is going to be this three by two matrix with first row 15 over to seven. Second row three halves 16 and third row five halves negative six. Okay. This is going to correspond with answer D. That's it for this one. Thanks everyone for watching. See you in the next video.

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. 3X + 2A = B

Key Concepts

Matrix Addition and Scalar Multiplication

Solving Matrix Equations

Matrix Dimensions and Compatibility

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