Video Transcript
If the matrix 𝐴 is equal to
negative one, zero, zero, six, eight, zero, six, five, three, which of the following
is true? (A) The matrix 𝐴 is an identity
matrix. (B) The matrix 𝐴 is an upper
triangle matrix. (C) The matrix 𝐴 is a lower
triangle matrix. (D) The matrix 𝐴 is a diagonal
matrix. Or (E) the matrix 𝐴 is a zero
matrix.
Let’s go through each of our types
of matrix and recall what they actually mean. An identity matrix is a square
matrix where all the elements is zero except for the elements on the leading
diagonal, which are equal to one. A three-by-three identity matrix,
for instance, would look as shown with the elements one, zero, zero, zero, one,
zero, zero, zero, one. Next, we have a zero matrix,
sometimes known as a null matrix. This is a square matrix whose
elements are all equal to zero. Now, in fact, if we compare the
matrix 𝐴 with either of these definitions, we see that it cannot be an identity
matrix, so we disregard (A). Nor can it be a zero matrix. And so, we’re going to disregard
(E).
Let’s now consider options (B),
(C), and (D). Now, a triangle matrix is a special
type of square matrix. An upper triangle matrix is a
square matrix where all the entries below the main diagonal are zero. Then we say it’s a lower triangle
matrix if the entries above the main diagonal are zero. We recall that the main diagonal or
the leading diagonal is this one. And so, do either of these two
definitions hold? Well, yes, they do. The elements that sit above this
diagonal are all equal to zero. And so this must be a lower
triangle matrix. And so the answer is (C).
We will just double check the
definition of (D). What does it mean for a matrix to
be diagonal? Well, for a matrix to be diagonal,
it must have entries below the main diagonal that are zero and above the main
diagonal that are zero. Then the entries on the actual
diagonal itself are not equal to zero. Of course, if we look carefully, we
see this isn’t the case with matrix 𝐴. The elements that sit below the
leading diagonal are not equal to zero. And so the answer is (C). The matrix 𝐴 is a lower triangle
matrix.
