Question Video: Identifying Symmetric Matrices Mathematics

Which of the following matrices is symmetric? [A] [−4, −4, 2 and −4, 8, −2 and 8, −2, 3] [B] [−2, 3, 5 and 3, −4, −3 and 5, −3, −7] [C] [−2, 4 , −3 and 4, −8, −8 and 9, −3, −2] [D] [−3, −5, −1 and −6, 6, 5 and −7, 7, −8]

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Video Transcript

Which of the following matrices is symmetric? (A) The matrix negative four, negative four, two, negative four, eight, negative two, eight, negative two, three. (B) The matrix negative two, three, five, three, negative four, negative three, five, negative three, negative seven. (C) The matrix negative two, four, negative three, four, negative eight, negative eight, nine, negative three, negative two. Or (D) the matrix negative three, negative five, negative one, negative six, six, five, negative seven, seven, negative eight.

Let’s begin by recalling what it means for a matrix to be symmetric. A matrix is symmetric if it is equal to its transpose, the matrix formed by swapping the rows and columns of the original matrix around. A matrix can only be symmetric if it is a square matrix, as the number of rows and columns must be the same so that both the matrix and its transpose are of the same order. Each of the matrices we’ve been given as options have three rows and three columns. So they’re each square matrices of order three by three. And so, at this stage, it’s possible that any of them could be symmetric matrices.

To answer this question, let’s find the transpose of each matrix. Remember, we do this by swapping the rows and columns around. So we may find it helpful to write each column of the original matrix in a different color. The first column of this matrix becomes the first row in its transpose. The second column becomes the second row. And then the third column becomes the third row. So the transpose of matrix 𝐴 is the matrix negative four, negative four, eight, negative four, eight, negative two, two, negative two, three.

Now, in order for matrices to be equal, it must be the case that every element in one matrix is equal to the corresponding element in the other. We can see that this is true for certain elements in the matrices 𝐴 and 𝐴 transpose. However, there are some elements for which this is not the case. So the matrix 𝐴 transpose is not equal to the matrix 𝐴, and so it is not a symmetric matrix.

Let’s now consider matrix 𝐵. And once again, we write each column in a different color. We can fill in the first row in the transpose matrix, then the second, and finally the third. And we find that the transpose of matrix 𝐵 is the matrix negative two, three, five, three, negative four, negative three, five, negative three, negative seven. This time, when we compare the two matrices, we can see that every element in the matrix 𝐵 is equal to the corresponding element in the matrix 𝐵 transpose. So the matrix 𝐵 is equal to its own transpose, and hence the matrix 𝐵 is symmetric.

We also need to check matrices 𝐶 and 𝐷, however. The matrix 𝐶 transpose is equal to negative two, four, nine, four, negative eight, negative three, negative three, negative eight, negative two. This time, we see that whilst there are certain elements that are the same in the matrix 𝐶 and its transpose, this isn’t true for every element. And so the matrix 𝐶 transpose is not equal to the matrix 𝐶, and so it is not symmetric.

Finally, we find the transpose of matrix 𝐷, which is equal to the matrix negative three, negative six, negative seven, negative five, six, seven, negative one, five, negative eight. This time, we find that the only elements that are the same in both the transpose matrix and the original matrix are those on the leading diagonal. All other elements are different in the two matrices. And so the matrix 𝐷 is not a symmetric matrix.

We found that the only one of the four matrices which is symmetric, so it is equal to its own transpose, is matrix 𝐵, the matrix negative two, three, five, three, negative four, negative three, five, negative three, negative seven.

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