If 𝐴 minus 𝐴 transpose is equal to a zero matrix, then 𝐴 is a blank matrix. Option (A) skew symmetric, option (B) symmetric, option (C) row, or option (D) column.
In this question, we’re given a property about the matrix 𝐴. We’re told 𝐴 minus its transpose is equal to a zero matrix. Given that this property is true, we need to determine what type of matrix matrix 𝐴 must be. And we’re given four possible options. And one way of answering this question is to recall exactly what our four answers mean.
Let’s start with what we mean by a skew symmetric matrix. We call a matrix 𝐵 skew symmetric if 𝐵 transpose is equal to negative 𝐵. In other words, taking the transpose of our matrix is the same as multiplying it by negative one. And there is a second way of thinking about this. In our matrix equation, we could add 𝐵 to both sides of our equation. This would give us the equivalent statement 𝐵 plus 𝐵 transpose should be equal to the zero matrix of the same order as 𝐵. And this is almost exactly the form of the property given to us in the question. However, instead of adding 𝐵 transpose, we need to subtract 𝐵 transpose.
Let’s instead recall what we mean by a symmetric matrix. We call a matrix 𝐵 symmetric if it’s equal to its own transpose. And once again, we can find an equivalent statement for this property by subtracting 𝐵 transpose from both sides of the equation. This gives us if 𝐵 minus 𝐵 transpose is equal to zero, then 𝐵 must be a symmetric matrix. And we can see this is exactly the property given to us in the question with matrix 𝐴. One way of seeing this is to add 𝐴 transpose to both sides of our matrix equation. We get that 𝐴 is equal to its own transpose. So, 𝐴 has to be a symmetric matrix.
Therefore, we know option (B) is true; 𝐴 has to be a symmetric matrix. However, this isn’t the only way of answering this question. Instead of starting with the properties we want to show, let’s start with the matrix equation we’re given. We’re told that 𝐴 minus its transpose is equal to the zero matrix. The first thing we need to notice here is when we take the transpose of a matrix, we switch the number of rows and columns around. But we can still subtract these two matrices, and we can only do this if these matrices are of the same order. So the order of 𝐴 and 𝐴 transpose must be equal. In other words, 𝐴 has to be a square matrix. And, of course, if 𝐴 is a square matrix, then our zero matrix is also a square matrix.
So let’s try and find the exact properties matrix 𝐴 must have. To do this, we need an expression for the elements in row 𝑖 and column 𝑗 of 𝐴. Let’s call this 𝑎 𝑖𝑗. But we can then use this and our matrix equation to come up with a new equation. First, recall when we take the transpose of matrix 𝐴, we need to switch the rows and columns around of matrix 𝐴. In other words, the element in row 𝑖 and column 𝑗 of 𝐴 transpose is going to be 𝑎 𝑗𝑖. All we do is switch 𝑖 and 𝑗 around. And, of course, when we subtract two matrices of the same order, we just need to subtract the elements in the same row and column. Finally, we know that this is equal to the zero matrix. And we know in row 𝑖 and column 𝑗 of the zero matrix we will always have zero. This is because every element in the zero matrix is equal to zero.
Now, instead of subtracting our two matrices, we can combine this into one matrix with element in row 𝑖 column 𝑗, 𝑎 𝑖𝑗 minus 𝑎 𝑗𝑖. Of course, all we’re saying here is we’re subtracting the corresponding entries of matrix 𝐴 and 𝐴 transpose. But for these two matrices to be equal, every single entry has to be equal and they must be of the same order. To guarantee that this equation makes sense, we did need 𝐴 to be a square matrix, and then they’ll be of the same order provided we chose our zero matrix of the same dimensions as matrix 𝐴.
Therefore, for these two matrices to be equal, we must have for all 𝑖 and 𝑗, 𝑎 𝑖𝑗 minus 𝑎 𝑗𝑖 is equal to zero. In other words, 𝑎 𝑖𝑗 is equal to 𝑎 𝑗𝑖 for all values of 𝑖 and 𝑗. And it might be worth pointing out here when we say all values of 𝑖 and 𝑗, we mean all values of 𝑖 and 𝑗 between one and the number of rows and columns of our matrix. And if 𝑎 𝑖𝑗 is exactly equal to 𝑎 𝑗𝑖 for all values of 𝑖 and 𝑗, this is exactly the same as saying that 𝐴 is equal to its own transpose because these two matrices are of the same order and we’ve just shown all of their corresponding entries are equal. And, of course, this is exactly the same as saying that 𝐴 is a symmetric matrix.
Now it’s worth pointing out the second method was a lot more complicated than the first method. However, it’s not always as easy to see how we can manipulate our property into one of our definitions. So sometimes we will need to work more closely with our definition of the matrix 𝐴. And the final thing we’ll point out is option (C) and option (D) can’t be correct. First, if a matrix only has one row, then we say that it’s a row matrix. And if a matrix only has one column, then we say it’s a column matrix.
But remember, right at the start of our property, we know it is that 𝐴 had to be a square matrix because we’re told we’re allowed to subtract 𝐴 from its own transpose. And transposing a matrix will switch the number of rows and columns around. So this only makes sense if 𝐴 has the same number of rows and columns. So 𝐴 can’t be a general row matrix, and it also can’t be a general column matrix because then the only time we can subtract 𝐴 from its own transpose is when it’s a one-by-one square matrix. Therefore, we were able to show if 𝐴 minus its own transpose is equal to the zero matrix, then 𝐴 must be a symmetric matrix, which was given as option (B).