Question Video: Finding Equivalent Matrices Using Scalar Multiplication of Matrices

Let 𝑍 be a 2 Γ— 3 matrix whose entries are all zero. If 𝐴 is any 2 Γ— 3 matrix, which of the following is equivalent to 5𝐴 βˆ’ 3𝑍? [A] 2𝑍𝐴 [B] βˆ’2𝐴𝑍 [C] βˆ’3𝑍 [D] 5𝐴 [E] 2𝐴

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Video Transcript

Let 𝑍 be a two-by-three matrix whose entries are all zero. If 𝐴 is any two-by-three matrix, which of the following is equivalent to five 𝐴 minus three 𝑍? Option (A) two times 𝑍 times 𝐴, option (B) negative two times 𝐴 times 𝑍, option (C) negative three 𝑍, option (D) five 𝐴, or option (E) two 𝐴.

In this question, we’re given an expression in terms of two matrices, and we need to determine which of five options is this expression equal to. And in fact, there’s a lot of different ways of answering this question. However, there’s one important thing we do need to notice. Both matrix 𝑍 and matrix 𝐴 are two-by-three matrices. This means the number of rows on one matrix and the number of columns on the other matrix are never equal. And when this happens, this means we can’t multiply 𝐴 by 𝑍, and we can’t multiply 𝑍 by 𝐴. So options (A) and (B) can’t be correct.

Let’s now see what we can do with the expression we’re given. Let’s start by recalling that 𝑍 is a two-by-three matrix, where every entry is equal to zero. Now we could write this out in terms of matrices. However, we can also write this as the two-by-three zero matrix, zero, two, three. Let’s now write out matrix 𝐴 and the zero matrix in full. To do this, we need to recall a matrix of order two by three will have two rows and three columns. We’ll also call the entry in matrix 𝐴 in row 𝑖, column 𝑗 π‘Ž 𝑖𝑗. This gives us the following expression.

We can now simplify this expression either by using our definition for scalar multiplication of a matrix or by using the fact that for any number π‘˜, π‘˜ multiplied by the π‘š-by-𝑛 zero matrix is just equal to the π‘š-by-𝑛 zero matrix. This gives us that three 𝑍 is just equal to the two-by-three zero matrix. Now we see we’re subtracting the zero matrix from our matrix five 𝐴. There’s a few different things we could do. For example, we could use our definition of scalar multiplication to bring five inside of our matrix. Then we could use our definition of matrix subtraction to answer this question.

However, this is not necessary because we’re subtracting the zero matrix of the same order. And when we do this, we subtract zero from every entry inside of our matrix. Of course, subtracting zero is not going to change any of the values, so this is just equal to five 𝐴. And this gives us that option (D) is the correct answer. It’s also worth pointing out we could check that option (C) and option (E) are not correct in all scenarios. For example, if we set 𝐴 equal to the two-by-three matrix where all entries are one, then we’ve already shown that five 𝐴 minus three 𝑍 should be equal to five 𝐴. And, of course, five 𝐴 is every entry in 𝐴 multiplied by five. It’s the two-by-three matrix where every entry is five.

This is, of course, not equal to two 𝐴, since this would be the two-by-three matrix where every entry is two, and it’s also not equal to the matrix negative three, 𝑍 since we’ve already shown that this will be equal to the two-by-three zero matrix. Therefore, we were able to show the only correct option is option (D); five 𝐴 minus three 𝑍 will be equal to five 𝐴.

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