Video Transcript
Consider matrix 𝐴 is equal to 𝑎 sub 𝑖𝑗, where 𝑖 is equal to one, two, and three and 𝑗 is equal to one and two. True or False: This matrix is a square matrix.
In this question, we’re given a matrix 𝐴 where the entry in row 𝑖, column 𝑗 is given by 𝑎 sub 𝑖𝑗. And we’re told that 𝑖 can vary from one to three and 𝑗 can vary from one to two. We need to use this to determine if matrix 𝐴 is a square matrix. To do this, let’s start by recalling what we mean by a square matrix. This is when a matrix has the same number of rows as its columns. And in fact, this means we can answer our question directly.
Since 𝑎 sub 𝑖𝑗 is the entry in row 𝑖, column 𝑗 of our matrix and our values of 𝑖 range from one to three and the values of 𝑗 range from one to two, our matrix must have three rows and two columns. Since these values are not equal, we can conclude 𝐴 is not a square matrix, which means we can just conclude that the answer is false. 𝐴 is not a square matrix.
However, it can be useful to see exactly why this is true. So, let’s write out the matrix 𝐴 by using its definition. First, we know 𝐴 is a matrix and 𝑎 sub 𝑖𝑗 tells us the entry in row 𝑖, column 𝑗 of our matrix. So, 𝑎 sub one one is the entry in row one, column one of our matrix. Similarly, 𝑎 sub one two is the entry in row one, column two. And we can’t go any further because in the question we’re told that values of 𝑗 only range between one and two. So, instead, we need to increase the value of 𝑖, which means we move on to a new row. The entry in row two, column one is 𝑎 sub two one, and the entry in row two, column two is 𝑎 sub two two. We can then do exactly the same for the third row. And then we know our values of 𝑖 can only range from one to three. So we can’t add any more rows to our matrix. This then gives us our matrix 𝐴. And this confirms that 𝐴 is not a square matrix. It has three rows and two columns. Therefore, the answer is false.
