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. 4A + 3B = - 2X

Accepted Answer

Hey everyone welcome back in this problem. We are asked to consider the matrices A and B. That were given and were told to solve the matrix equation for X. And the matrix equation we're given is six A plus two, B is equal to negative three X. Yes, we have six a Plus two B is equal to -3. No. Were given an equation like this and we want to solve for X. One thing we can do is isolate for X right off the bat. Okay, this can make it easier in the long run, so let's go ahead and do that. So this is going to be six divided by negative three is negative two A. Okay, we're gonna get minus two thirds. B is equal to X. Okay, so we've just divided by -3. Like we would in a normal equation and this can just make it simpler. We can do all of the work up front and just get X directly. You don't have to do that if you don't want you can work it out and then compare terms. But we're gonna do it this way. Alright, so we're talking negative two times A. So we're gonna have negative two. The matrix A. Is a three by two. Matrix the first column is negative 610. The second column is negative four negative 83. Okay, and we're gonna subtract two thirds B B has first column negative 975 and second column negative 200. Alright, so we have some multiplying by a constant. And we have subtraction as well. Okay, when we have both of those things that we want to do is we want to do the multiplication first. Okay. Just like with normal equations, let's do the multiplication first. And when we're multiplying a matrix by a constant, that's going to be an element wise operation. Okay. So we're gonna take that constant. Bring it into the matrix and it's gonna multiply to every single element in the matrix. Ok. So applying this to the first matrix, what we're going to have Because we're gonna have negative two times negative six. Okay, we're gonna have negative two times one, negative two times zero. Okay, As our first call And in the second call and we're gonna get negative two times negative. four -2 times negative eight And -2 times three. Alright, so that's gonna be the first matrix and then we're subtracting and the second matrix is going to be two thirds times negative nine, two thirds times seven. And two thirds time starts. And the second column for the second matrix we get two thirds times negative, two, 2/3 time zero And 2/3 times zero. Alright, so we've dealt with the multiplication. Okay, let's just simplify within each of these matrices. And then all that's left to do is the subtraction. All right. So the first matrix, when we do the multiplication, we're gonna get 12 -20 in the first column eight. 16 -6 in the 2nd column. And we're subtracting the second matrix which is going to be okay. So we have two times negative nine divided by three. So we're gonna have negative 18 divided by three. That's going to be negative six. Okay, two thirds times seven. We're gonna get 14/3 and 10/3 for the first column, 3rd row, moving to the second column. We get negative 4/3 00. Alright, So we've simplified both of those. Now. We need to do the subtraction. We want to double check when we're subtracting matrices that the dimensions are the same. Okay. They must be the same in order to do a subtraction in this case they're both three by two matrices. So we're okay to go ahead. Subtraction is an element wise operation. Okay. So what that means is we're going to take an element from the first matrix? We're going to subtract the corresponding element from the second matrix. Okay. And that's going to go into our new matrix in the same position. So let's see that in practice. Okay. So what we're doing, we're taking the elements say from row one column one. In the first matrix. Ok. The corresponding element in the second matrix is in the same position. Row one column one. So it's negative six. Okay. We're gonna put it in the same position in our resulting matrix. So row one column one. And we're gonna do the subtraction. So we have 12 minus negative six. We're gonna do the same thing for the second row. First column in the first matrix we have negative two. In the same position of the second matrix we have 14/3. So we're gonna get negative two -14/3. Okay. Going again. Second row, first column in the resulting matrix and we can do the same for the rest of the term. So we have zero minus 13th. Okay, in the second column, eight minus negative four thirds 16 0 and - 0. Okay, so now we are down to one matrix like we wanted we just need to simplify inside of this matrix. Ok and you'll notice that the resulting matrix, remember the dimension when you subtract the dimension of the two matrices has to be the same. The dimension of the resulting matrix is going to be the same as that as well. Alright, so let's give ourselves a little bit more room here just to simplify this matrix. Okay, so we're gonna have 12 -6. This is going to be 18. A negative two minus 14/3. Well this is going to be negative 6/3 minus 14/3. Okay, just to get a common denominator. The final row first column is gonna be negative 10/3. Second column. First row we have eight minus negative four thirds. Okay, well this is going to be 24/3 plus 4/3. Okay. Getting a common denominator there in the second row. Second column we get 16 and bottom row, second column negative six. Alright, so working those last few out we got negative 20/3, negative 10/3 28. Over three 16 and -6. And so this is the resulting matrix when we do the operation and solve for X. Okay, so the matrix is a three by two matrix with first column 18, negative 20/3, negative 10/3. And second column 28/3, 16 and negative six. Okay. If we go back up to our possible answers, you'll see that this corresponds with answer A. That is not answer. A answer A is above. There we go. That's the one we were looking for answer A here. Alright, 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. 4A + 3B = - 2X

Key Concepts

Matrix Operations

Solving Matrix Equations

Scalar Multiplication and Division of Matrices

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