Video Transcript
Determine which of the following matrices is a lower triangular matrix. Option (A) the three-by-three matrix one, zero, zero, five, seven, zero, nine, five, six. Option (B) the three-by-three matrix one, five, nine, zero, seven, five, zero, zero, six. Option (C) the three-by-three matrix one, zero, nine, five, zero, five, nine, zero, six. Option (D) the three-by-three matrix zero, five, nine, zero, seven, five, zero, one, six. Or option (E) the three-by-three matrix one, five, nine, zero, zero, five, one, zero, six.
In this question, we’re given five possible matrices, and we need to determine which of these matrices is a lower triangular matrix. So to do this, we’re first going to need to recall what we mean by a lower triangular matrix. We recall that we call a matrix a lower triangular matrix if all of the entries above the main diagonal of our matrix are equal to zero. And remember when we say the main diagonal of a matrix, this is all of the entries of our matrix, which are in the same row number and column number. In other words, for matrix (A), we can take the entry in row one, column one; the entry in row two, column two; and the entry in row three, column three. This is the main diagonal of our matrix in option (A).
We need to check all of the entries above this main diagonal are equal to zero. So we need to check the three entries above this main diagonal are equal to zero. In fact, we can see that this is true. Therefore, the matrix in option (A) is lower triangular because all of the entries above the main diagonal are equal to zero. Now, we might want to stop here. However, we might as well check the rest of our options. Let’s check if the matrix in option (B) is a lower triangular matrix. First, the main diagonal of this matrix will be the entry in row one, column one; the entry in row two, column two; and the entry in row three, column three. For this to be a lower triangular matrix, all of the entries above this main diagonal have to be equal to zero, and we can see this is not true. In fact, none of these entries are equal to zero. So the matrix in option (B) is not a lower triangular matrix.
We can do the same for the matrix in option (C). We mark its main diagonal and then we check all of the entries above the main diagonal are equal to zero. Once again, we can see this is not true. Two of these entries are not equal to zero. So the matrix in option (C) is not a lower triangular matrix. We can do exactly the same thing for the matrix in option (D). We mark its main diagonal and check whether the entries above this main diagonal are equal to zero. And once again, we can see that not all of these entries are equal to zero. In fact, none of them are. Therefore, the matrix in option (D) is also not a lower triangular matrix.
Finally, we want to check matrix (E). We’ll start by marking its main diagonal. That will be the entry in row one, column one; the entry in row two, column two; and the entry in row three, column three. Finally, to check whether this is a lower triangular matrix, all we need to do is check whether all of the entries above the main diagonal are equal to zero. Of course, we can see that none of these are equal to zero. So the matrix in option (E) is also not a lower triangular matrix. Therefore, we were able to show, of the matrices listed, only the matrix in option (A) — which is the three-by-three matrix one, zero, zero, five, seven, zero, nine, five, six — is a lower triangular matrix.
