Let A = < b m a t r i x > and B =

Let $A = \begin{bmatrix} a_{11} & a_{12} \ a_{21} & a_{22} \end{bmatrix}$ and $B = \begin{bmatrix} b_{11} & b_{12} \ b_{21} & b_{22} \end{bmatrix}$, where all the elements are real numbers. Use these matrices to show that the statement is true for 2x2 matrices. A + B = B + A (commutative property)

Transcript

00:01 Hello, everybody.

00:03 In this problem, we're asked to prove that the addition of matrices follows the associated property, which basically means that i can add them in any order, add three matrices in any order that i want, and so i can group them in any order so i can add a and b first and then add c or i can add b and c first and then add a, and it still gives me the correct solution.

00:25 And so we want to show that when we add a to b plus c, so we do b plus c first over here on the left.

00:32 That's saying we're doing b plus c first.

00:35 And then we're going to add a.

00:36 And over here on the right, we're going to add a plus b first and then add c.

00:41 And we want to show that those make the same matrices.

00:44 Right.

00:44 So i've already set up the matrices here with their corresponding to like a11, a12, and so on.

00:53 I've already set that up with the parentheses.

00:56 And so we're going to do addition of these matrices.

00:59 So since we've got things in parentheses, we always do the things in parentheses first and quarter of the order of operations.

01:05 And so i'm going to do the thing in parentheses on the left side here.

01:08 Remember when we're adding matrices, we're adding the corresponding element.

01:12 And so it's the elements that are in the same spot.

01:15 Since all of these represent real numbers, so like b sub 1 or c sub 1, those represent real numbers.

01:22 And so those are placeholders.

01:23 And so we're adding b sub sub 1, sub 1, and c sub 1...

Question

Let $A = \begin{bmatrix} a_{11} & a_{12} \ a_{21} & a_{22} \end{bmatrix}$ and $B = \begin{bmatrix} b_{11} & b_{12} \ b_{21} & b_{22} \end{bmatrix}$, where all the elements are real numbers. Use these matrices to show that the statement is true for 2x2 matrices. A + B = B + A (commutative property)

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