MATH-242 - Linear Algebra Chapter 2 Take Home Worksheet 2.2...

MATH-242 - Linear Algebra Chapter 2 Take Home Worksheet 2.2 Properties of Matrix Operations 1. Let A, B, and C be matrices such that the operations indicated below are defined. Factor each of the following matrix expressions. (Remember that matrix multiplication is not commutative!) (a) AB + 2CB (b) 3AB + AC + A (c) A(B - C) + A(C + B) (d) AB - BA + AC - CA 2. TRUE/FALSE: Determine whether each of the following is true or false. (a) Let A and B be m × n matrices and C be an n × p matrix. If AC = BC, then A = B. (b) For n × n matrices A and B, the formula (A + B)(A - B) = A$^2$ - B$^2$ holds. (c) In general, multiplication of a matrix B on the left by A would produce the same matrix as would multiplication of a matrix B on the right by A.

Transcript

00:04 And in this question, we're given three madries, a, b, c, right? and mark each statement two forth, right? so let's look at this statement.

00:13 The first says, if a and b are two by two with columns like this, a and b are two by two with columns a1, a2, b1 with respect to the right.

00:26 The a, b, a, like this, you ask two feet in the blanks, right? so the statement, using the statement is correct, where you look at the basically product of ab, the product of ab is the product of ab, if the product means that, so basically if you look at the a that's given by, you know, a1, which is a column way.

00:50 So you have a1 and a2 way.

00:53 And the b will be b1, b2 columns as well, right? and that's b, right? you look at the product of a and b, and what would you actually get you would get actually four by four metrics as well right and and you know the first one you're looking the the first one you would actually get is is definitely not a one times b one a two times b2 right certain only this right so this this statement actually is false so i think the the statement is correct it's it's a false statement right so um so ab as i said it's going to be like this right so you have the definition matrix multiplication states that if a is m -m matrix, b is m -p matrix, right? and the ab will be a matrix, which is of size m and cross -p, right? so that would be the size, and that would be it.

01:56 And b, each column of ab is a linear combination of the columns b using weights from the corresponding columns a.

02:02 Each column of ab is a new combination of the columns b, using weights from the corresponding column of a.

02:13 Well, actually the ab, you can see that is actually going to be given by.

02:21 So each column of, each column of ab, right, with ab, and that will actually be given by.

02:29 So you have a look at a1 and a2, right, for example, and this b1 and b2, right...

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