If matrix 𝐴 has order of 𝑛 by 𝑛, where 𝑛 is greater than one, such that 𝑎 𝑖𝑗 equals two for 𝑖 equal to 𝑗 and 𝑎 𝑖𝑗 equals zero for 𝑖 not equal to 𝑗, which of the following is its type? (A) The identity matrix, (B) the zero matrix, (C) row matrix, (D) diagonal matrix, or (E) column matrix.
So, we can have a look at each of the options here in a moment, but let’s first go through the information given in the question. Matrix 𝐴 has order of 𝑛 by 𝑛. This means it has 𝑛 rows and 𝑛 columns, so the same number of rows as columns, meaning that matrix 𝐴 must be a square matrix. We also have that 𝑛 is greater than one, so matrix 𝐴 has more than one row and more than one column. We have that 𝑎 𝑖𝑗 is equal to two for 𝑖 equal to 𝑗, meaning that where we have the same row number and same column number, the entry is two. So, 𝑎 one one equals two, 𝑎 two two equals two, 𝑎 three three equals two, and so on. And we also have that 𝑎 𝑖𝑗 equals zero when 𝑖 is not equal to 𝑗. So, all the entries that are not two are going to be zero.
Okay, so let’s go through the options that we’ve been given here. Option (A) is that this is an identity matrix. Remember that the identity matrix is a very special matrix, with ones along the main diagonal and zeros everywhere else, such as the matrix one, zero, zero, zero, one, zero, zero, zero, one. But because we’ve been told that 𝑎 𝑖𝑗 is equal to two where 𝑖 is equal to 𝑗, meaning that the entries 𝑎 one one is two, 𝑎 two two is two, 𝑎 three three is two, and so on, these are all diagonal entries. So, the matrix in the question has twos along the diagonal from top left to bottom right. Therefore, it cannot be the identity matrix. Okay, so could this be a zero matrix?
Well, with a zero matrix, all of the entries are zero, like in the matrix zero, zero, zero, zero. But we know that the matrix in the question has twos along the main diagonal. Therefore, it cannot be a zero matrix. So, now, let’s consider whether matrix 𝐴 could be a row matrix. A row matrix is a matrix whose entries are arranged in one row. It can have any number of columns but only one row. But because the matrix in this question has order of 𝑛 by 𝑛 where 𝑛 is greater than one, it cannot have only one row. So, this is not a row matrix.
So, let’s see whether this matrix is a diagonal matrix. A diagonal matrix is a square matrix whose nondiagonal entries are zero. The entries along the main diagonal can either be constants or they can be zero. Well, we know the matrix in this question has entries of zero, which are not on the main diagonal. We also know because it’s an 𝑛-by-𝑛 matrix that it is a square matrix, and the entries along the diagonal are all two. So, matrix 𝐴 is a diagonal matrix.
Let’s just check the final option. Could it be a column matrix? A column matrix is a matrix whose entries are all arranged in one column. The matrix can have any number of rows, but only one column. But because we were told that this matrix has order of 𝑛 by 𝑛, where 𝑛 is greater than one, this cannot be a column matrix because matrix 𝐴 has more than one column.
Therefore, matrix 𝐴 is a diagonal matrix with twos along the main diagonal from top left to bottom right and zeros everywhere else.