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 𝑎 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 𝑘, 𝑘 multiplied by the 𝑚-by-𝑛 zero matrix is just equal
to the 𝑚-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.
