If the matrix 𝐴 equals one, zero, zero, zero, eight, zero, zero, zero, negative three, which of the following is true? (A) The matrix 𝐴 is a column matrix. (B) The matrix 𝐴 is a diagonal matrix. (C) The matrix 𝐴 is a zero matrix. (D) The matrix 𝐴 is a row matrix. Or (E) the matrix 𝐴 is an identity matrix.
Let’s begin this question by looking at the options we’ve been given. Option (A) says that the matrix 𝐴 is a column matrix. But what does this mean? Well, as the name suggests, it’s a matrix which is just one column, that is, a matrix of order 𝑚 by one. Remember, by order we just mean the size of the matrix. So we’re saying that a column matrix has 𝑚 number of rows and one column, such as the matrix five, zero, negative three.
A diagonal matrix is a square matrix whose nondiagonal entries are zero. The diagonal entries are the entries with the same row and column number. So any entry outside of that main diagonal is zero, such as the matrix four, zero, zero, one.
Option (C) is that this is a zero matrix. A zero matrix is a matrix where all of the entries are zero. An example would be the matrix zero, zero, zero, zero, zero, zero, zero, zero, zero.
Option (D) is that this is a row matrix. What we mean by that is that’s a matrix with order one by 𝑛. So it’s just one row. So an example would be the matrix negative three, zero, four, two.
And finally option (E) is that this is an identity matrix. That is a matrix who has ones along the main diagonal and zeros everywhere else, such as the two-by-two identity matrix one, zero, zero, one.
So we’ve got to decide which of these is true for our matrix. One observation that we can make about our matrix is that it has three rows and three columns. Therefore, this is a three-by-three matrix. Another thing we can say is that because it has three rows and three columns, the same number of rows as columns, it’s a square matrix. So we can see that this matrix is not just one column or just one row. So we can rule out options (A) and (D).
So the options we have left is that 𝐴 is a diagonal matrix, 𝐴 is a zero matrix, or 𝐴 is an identity matrix. We can see straightaway that matrix 𝐴 is not the zero matrix, because with the zero matrix all of the entries are zero. So this is not a zero matrix. We can also see that this is not the identity matrix because the identity matrix has to have only ones along the main diagonal. The main diagonal entries will be the entries from top left to bottom right. So that’s one, eight, and negative three. So, clearly, these are not one. Therefore, matrix 𝐴 cannot be an identity matrix.
So we’re left with matrix 𝐴 being a diagonal matrix. But let’s just double-check. We said that a diagonal matrix is a square matrix whose nondiagonal entries are zero. Well, we already decided that this was a square matrix. And we can see that any of the entries that are not in the main diagonal are zero. Therefore, the answer is that the matrix 𝐴 is a diagonal matrix.