symmetric matrix is headquarters in which each element \(a_{ij}\) is equal to the element \(a_{ji}\) for all values of i and j. Consequently, every symmetric matrix is equal to its transpose. It is also worth mentioning that every symmetric matrix is square and that the main diagonal acts as an axis of symmetry.
Read too:Matrix addition and subtraction — how to calculate?
Abstract about symmetric matrix
In a symmetric matrix, \(a_{ij}=a_{ji}\) for all i and j.
Every symmetric matrix is square.
Every symmetric matrix is equal to its transpose.
The elements of a symmetric matrix are symmetric about the main diagonal.
While in the symmetric matrix \(a_{ij}=a_{ji}\) for all i and j; in an antisymmetric matrix, \(a_{ij}=-a_{ji}\) for all i and j.
What is a symmetric matrix?
A symmetric matrix is a square matrix where \(\mathbf{a_{ij}=a_{ji}}\) for every i and every j. This means that \(a_{12}=a_{21},a_{23}=a_{32},a_{13}=a_{13}\), and so on, for all possible values of i and j. Remember that the possible values of i correspond to the rows of the matrix and the possible values of j correspond to the columns of the matrix.
Examples of symmetric matrices
\(\begin{bmatrix} 5 & 9 \\ 9 & 3 \\ \end{bmatrix}\), \(\begin{bmatrix} -2 & 1 & 7 \\ 1 & 0 & 3 \\ 7 & 3 & 8 \\ \end{bmatrix}\), \(\begin{bmatrix} a & b & c \\ b & d & e \\ c & e & f \\ \end{bmatrix}\)
Examples of non-symmetric matrices (consider \(\mathbf{b≠g}\))
\(\begin{bmatrix} 5 & 8 \\ 9 & 3 \\ \end{bmatrix}\), \(\begin{bmatrix} -2 & 1 & 7 \\ 1 & 0 & 3 \\ 4 & 3 & 8 \\ \end{bmatrix}\), \(\begin{bmatrix} a & g & c \\ b & d & e \\ c & e & f \\ \end{bmatrix}\)
Important: To say that a matrix is not symmetric means to show that \(a_{ij}≠a_{ji}\) for at least some i and j (which we can see by comparing the previous examples). This is different from the antisymmetric matrix concept, which we will see later.
What are the properties of the symmetric matrix?
Every symmetric matrix is square
Note that the definition of a symmetric matrix is based on square matrices. Thus, every symmetric matrix has the same number of rows as the number of columns.
Every symmetric matrix is equal to its transpose
If A is a matrix, its transposed (\(A^T\)) is defined as the matrix whose rows are the columns of A and whose columns are the rows of A. So, if A is a symmetric matrix, we have \(A=A^T\).
In the symmetric matrix, the elements are “reflected” with respect to the main diagonal
As \(a_{ij}=a_{ji}\) in a symmetric matrix, the elements above the main diagonal are “reflections” of the elements below of the diagonal (or vice versa) in relation to the diagonal, so that the main diagonal acts as an axis of symmetry.
What are the differences between the symmetric matrix and the antisymmetric matrix?
If A is a symmetric matrix, then \(a_{ij}=a_{ji}\) for all i and all j, as we studied. In the case of the antisymmetric matrix, the situation is different. If B is an antisymmetric matrix, then \(\mathbf{b_{ij}=-b_{ji}}\) for every i and every j.
Note that this results in \(b_{11}=b_{22}=b_{33}=⋯=b_{nn}=0\), that is, the main diagonal elements are zero. A consequence of this is that the transpose of an antisymmetric matrix is equal to its opposite, that is, if B is an antisymmetric matrix, then \(B^T=-B\).
Examples of antisymmetric matrices
\(\begin{bmatrix} 0 & -2 \\ 2 & 0 \\ \end{bmatrix}\), \(\begin{bmatrix} 0 & 5 & -1 \\ -5 & 0 & 4 \\ 1 & -4 & 0 \\ \end{bmatrix}\), \(\begin{bmatrix} 0 & -m & x \\ m & 0 & -y \\ -x & y & 0 \\ \end{bmatrix}\)
See too: Identity matrix — the matrix in which the main diagonal elements are equal to 1 and the remaining elements are equal to 0
Solved exercises on symmetric matrix
question 1
(Unicentro)
if the matrix \(\begin{bmatrix} 1 & x & y-1 \\ y-1 & 0 & x+5 \\ x & 7 & -1 \\ \end{bmatrix}\) is symmetric, so the value of xy is:
A) 6
B) 4
C) 2
D) 1
E) -6
Resolution:
Alternative A
If the given matrix is symmetric, then the elements in symmetrical positions are equal (\(a_{ij}=a_{ji}\)). Therefore, we have to:
\(x = y - 1\)
\(x + 5 = 7\)
Replacing the first equation in the second, we conclude that \(y=3\), soon:
\(x=2\) It is \(xy=6\)
question 2
(UFSM) Knowing that the matrix \(\begin{bmatrix} Y & 36 & -7 \\ x^2 & 0 & 5x \\ 4-y & -30 & 3 \\ \end{bmatrix}\) is equal to its transpose, the value of \(2x+y\) é:
A) -23
B) -11
C) -1
D) 11
E) 23
Resolution:
Alternative C
Since the given matrix is equal to its transpose, then it is a symmetric matrix. Thus, elements in symmetrical positions are equal (\(a_{ij}=a_{ji}\)), i.e:
\(x^2=36\)
\(4-y=-7\)
\(-30=5x\)
By the first equation, x=-6 or x=6. By the third equation, we get the correct answer: x= -6. By the second equation, y=11.
Soon:
\(2x+y=2.(-6)+11=-1\)
By Maria Luiza Alves Rizzo
Math teacher
Source: Brazil School - https://brasilescola.uol.com.br/matematica/matriz-simetrica.htm