Video Transcript
Which of the following matrices is skew-symmetric? (A) The matrix 10, negative nine, negative two, negative nine, negative two, negative five, negative two, negative five, four. (B) The matrix three, negative five, negative two, five, negative three, one, two, negative one, three. (C) The matrix zero, negative three, negative five, three, zero, 10, five negative 10, zero. Or (D) the matrix zero, negative one, negative nine, one, zero, negative six, nine, six, seven.
Let’s begin by recalling what it means for a matrix to be skew-symmetric. A matrix is skew-symmetric if its transpose is equal to its negative. This is only possible for square matrices because it must be the case that the matrix and its transpose have the same dimensions. A property of skew-symmetric matrices is that the element 𝑎 sub 𝑗 sub 𝑖 must be equal to negative the element a sub 𝑖 sub 𝑗 for all 𝑖 and 𝑗. The four matrices we’ve been given as options each have three rows and three columns. So, they are each square matrices of order three by three. To find which matrix is skew-symmetric, let’s find the transpose of each.
We do this by swapping the rows and columns of each matrix around. So, the first column of each matrix will become the first row in its transpose. Let’s look at option (A) first of all. We may find it helpful to write each column in a different color. In the transpose matrix, the first column becomes the first row, the second column becomes the second row, and the third column becomes the third row. So, for option (A), the transpose matrix is equal to the matrix 10, negative nine, negative two, negative nine, negative two, negative five, negative two, negative five, four.
Now, in fact, this matrix is exactly equal to the original matrix. Every element in the transpose matrix is equal to the element in the same position in the matrix itself. This means that matrix 𝐴 has the property 𝐴 transpose equals 𝐴 and is therefore what is known as a symmetric matrix. We’re looking for a skew-symmetric matrix though. So, let’s consider matrix 𝐵.
Once again, we can write each column in a different color to help us when finding the transpose. Swapping the rows and columns around, we find that the transpose of matrix 𝐵 is the matrix three, five, two, negative five, negative three, negative one, negative two, one, three. Now, remember we’re looking for each element in the transpose matrix to be the negative of the corresponding element in the original matrix in order for the matrix to be skew-symmetric. This is true for certain elements, but it isn’t true for every element. The elements on the leading diagonal are, in fact, the same in both the original matrix and its transpose. So, it’s not the case that 𝐵 transpose equals negative 𝐵.
Let’s now consider option (C). Finding the transpose of this matrix gives the matrix zero, three, five, negative three, zero, negative 10, negative five, 10, zero. Now if we factor this matrix by negative one, we obtain negative the matrix zero, negative three, negative five, three, zero, 10, five, negative 10, zero. That’s the exact negative of the original matrix. So, we found that 𝐶 transpose is equal to negative 𝐶, which means that this matrix is skew-symmetric.
We think we found the answer then, but we must check option (D) as it could also be skew- symmetric. Once again, we’ll use colors for the different columns of matrix 𝐷 to help us when finding the transpose, which is equal to the matrix zero, one, nine, negative one, zero, six, negative nine, negative six, seven. This time, we find that each element in the transpose matrix is equal to the negative of the corresponding element in the original matrix with one exception. The element in the third row and third column is the same in both the original and the transpose matrices; it’s not the negative of itself. And so, matrix 𝐷 is not skew-symmetric.
In fact, options (B) and (D) reveal something helpful, which is that a matrix can only be skew-symmetric if all of the elements on its leading diagonal are equal to zero. When we find the transpose of a square matrix, the elements on the leading diagonal do not change. And so, the only way they can be equal to their own negative is if they are equal to zero. We can see that in option (C) the three elements on the leading diagonal are indeed all equal to zero, which isn’t the case for any of the other matrices. This wouldn’t be enough for us to conclude that option (C) was the correct answer, but it would be enough for us to rule out each of the others.
Of the four options given, the only matrix that is skew-symmetric is matrix 𝐶, the matrix zero, negative three, negative five, three, zero, 10, five, negative 10, zero.
