Question Video: Determining the Type of Matrix Given Mathematics

If the matrix 𝐴 = [βˆ’1, 0, 0 and 6, 8, 0 and 6, 5, 3], 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. [E] The matrix 𝐴 is a zero matrix.

02:30

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.

Nagwa uses cookies to ensure you get the best experience on our website. Learn more about our Privacy Policy.