Find the indicated entry of the matrix if it is defined a...

Find the indicated entry of the matrix, if it is defined. $$A = \begin{pmatrix} 1 & 3 & 1 \\ 2 & -1 & 4 \end{pmatrix}$$ (a) $$a_{2,1}$$ (b) $$a_{1,2}$$ (c) $$a_{2,2}$$ (d) $$a_{3,1}$$

Transcript

00:01 Okay, so here, well, we have both a and b are three by three matrices.

00:08 So a, b, and b are both possible.

00:13 So here, while the first row of ab is equal to the first row of matrix a times b.

00:22 So we have the first row of matrix a.

00:24 So we have 3 -7 -7 times b.

00:31 So times six, negative two, four, zero, one, three, seven, seven, five.

00:44 Okay.

00:48 Okay, so what do we get here? well, we do basically a dot prime.

00:54 We do three times six plus negative two times zero plus seven times seven.

01:05 We do three times negative two plus negative two times one plus seven times seven right we take each row multiplying it by the given column to get so first um we get three times six plus negative two times zero plus seven times seven that gives us 67 and then we get three times negative two plus negative 2 times 1 plus 7 times 7 gives us 41 and then we get 3 times 4 plus negative 2 times 3 plus 7 times 5 gives us 41 as well.

01:45 So indicated row vector here is given right here.

01:53 All right.

01:55 For part b, well, we look to find the third row of a.

02:03 So the third row of matrix a times b gives us the third row of matrix a, so we have 049 times b, times six, negative two, four, zero, one, three, seven, seven, five.

02:31 All right.

02:35 We do this.

02:35 We do zero times six, plus four times zero plus nine times seven gives us 63.

02:43 And then we do zero times two plus four times one plus nine times seven gives us 67.

02:53 And then we do nine, um, then we do zero times four plus four times three plus nine times five gives us 57.

03:10 All right, so there we have it.

03:15 All right, so this is...

03:16 And then for c, well, here we find the second column of ab is equal to a times the second column of b.

03:39 So, okay, so here we have matrix a.

03:43 So we have a, which is 3, negative 2, 7, 6, 5, 4, 0, 4, 9.

03:57 Okay, so we have a times the second column of b.

04:02 So the second column of b is negative 2, 1, 7.

04:09 All right.

04:11 So this is going to give us, while we get 3 times negative 2, plus two times one plus seven times seven gives us 41.

04:30 So we get 41.

04:31 And then we get six times negative two plus five times one plus four times seven gives us 21.

04:41 And then we get zero times negative two plus four times one plus nine times seven.

04:50 Gives us 67.

04:53 All right...

Question

Find the indicated entry of the matrix, if it is defined. $$A = \begin{pmatrix} 1 & 3 & 1 \\ 2 & -1 & 4 \end{pmatrix}$$ (a) $$a_{2,1}$$ (b) $$a_{1,2}$$ (c) $$a_{2,2}$$ (d) $$a_{3,1}$$

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