Question

If A is any matrix, then the product AA is defined only when A is a matrix of order \[m\times n\] where:
(a) m > n
(b) m < n
(c) m = n
(d) \[m\le n\]

Answer 1

Hint: Matrix multiplication between any two matrices are valid when the number of columns of the first matrix is equal to the number of rows of the second matrix. Here we will check when the multiplication of Matrix A with A is valid.

Complete step-by-step answer:
Here, A is any matrix. Let A have m number of rows and n number of columns, i.e. \[{{\left( A \right)}_{m\times n}}.\] So,
\[AA={{\left( A \right)}_{m\times n}}{{\left( A \right)}_{m\times n}}\]
We know that two matrices can be multiplied with each other if and only if the number of columns in the first matrix in the multiplication operation is equal to the number of rows in the second matrix. Therefore, in \[{{\left( A \right)}_{m\times n}}{{\left( A \right)}_{m\times n}},\] the column n of the first matrix should be equal to the row m of the second matrix. This implies that the value of m must be equal to the value of n that is m = n. Now because both the matrices are the same, that is, A itself, therefore it doesn't matter which matrix of the two is multiplied first.
Therefore, the number of rows should be equal to the number of columns, i.e, m should be equal to n. Therefore, m = n.
Hence, the option (c) is the right answer.

Note: If there are two matrices and both their orders are \[2\times 3,\] we cannot find the product of these 2 matrices as the number of columns of the first matrix is not equal to the number of rows of the second matrix. Also, it is unsafe to assume that the matrix multiplied by itself is possible as the number of rows and columns may or may not be the same in it.