What is Symmetric and Skew Symmetric Matrix?
Matrix: Matrix in mathematics is defined as an array of numbers arranged in a rectangular fashion and divided between rows and columns. It contains all the numbers arranged in square brackets. The operation of matrices is a very important topic in mathematics for the board exams as well as for the engineering entrance exams.
What is a Symmetric Matrix?
A square matrix that is equal to its transpose is known as a symmetric matrix.
Only square matrices are symmetric because only equal matrices have equal dimensions.
A matrix A with nn dimensions is said to be skew-symmetric if and only if
aij = aji for all i, j such that 1≤n, j≤n.
Suppose A is a matrix, then if the transpose of matrix A = AT is equal, it is a symmetric matrix.
Symmetric matrix example,
A = \[\begin{bmatrix}1 & 1 & -1\\ 1 & 2 & 0\\ -1 & 0 & 5\end{bmatrix}\]
The transpose of A (AT) = \[\begin{bmatrix}1 & 1 & -1\\ 1 & 2 & 0\\ -1 & 0 & 5\end{bmatrix}\]
Since, A= AT matrix A is a symmetric matrix.
NOTE: If any diagonal matrix is equal to the transpose of the matrix, such matrices are automatically symmetric.
Before We Move Further, Let Us Know About Some Important Terms!
A square matrix is a matrix where the number of columns is equal to the number of rows.
Here, m = The number of rows
n = The number of columns
Skew Symmetric Matrix Definition
A square matrix is said to be skew-symmetric if the transpose of the matrix equals its negative.
A matrix A with nn dimensions is said to be skew-symmetric if and only if
aij = -aji for all i, j such that 1≤n, j≤n.
Suppose A is a matrix, then if the transpose of matrix A, AT =- A is equal then it is a skew-symmetric matrix.
First, let us know how to find the Transverse of a Matrix
Transpose of a Matrix (AT)
We find the transpose of a matrix by interchanging the rows and columns of the original matrix. Suppose the original matrix is denoted by n×m, the transpose of the matrix will be m×n.
Let us take an example,
If A= \[\begin{bmatrix}1 & 2\\ 3 & 4\end{bmatrix}\], then let us calculate the transpose of the matrix A.
Here, the first row becomes the first column and the second row becomes the second column.
AT= \[\begin{bmatrix}1 & 3\\ 2 & 4\end{bmatrix}\]
Here we see that A AT.
If A = \[\begin{bmatrix}1 & 1 & -1\\ 1 & 2 & 0\\ -1 & 0 & 5\end{bmatrix}\], then let us calculate the transpose of the matrix A.
Here, the first row becomes the first column, the second row becomes the second column and the third row becomes the third column.
AT = \[\begin{bmatrix}1 & 1 & -1\\ 1 & 2 & 0\\ -1 & 0 & 5\end{bmatrix}\]
Here, we see that A = AT
How to check Whether a Matrix is Symmetric or Not?
Step 1- Find the transpose of the matrix.
Step 2- Check if the transpose of the matrix is equal to the original matrix.
Step 3- If the transpose matrix and the original matrix are equal, then the matrix is symmetric.
Example 1
A = \[\begin{bmatrix}0 & 2 & -45\\ -2 & 0 & -4\\ 45 & 4 & 0\end{bmatrix}\]
- A= \[\begin{bmatrix}0 & -2 & 45\\ 2 & 0 & 4\\ -45 & -4 & 0\end{bmatrix}\]
= AT
Since, AT=-A matrix A is a skew-symmetric matrix
Example 2
P = \[\begin{bmatrix}0 & -5\\ 5 & 0\end{bmatrix}\]
-P = \[\begin{bmatrix}0 & 5\\ -5 & 0\end{bmatrix}\]
= PT
Since, PT=-P matrix A is a skew-symmetric matrix.
Conditions for Symmetric and Skew Symmetric Matrix
Here, i = Row entry
j = Column entry
How to check whether a Matrix is Skew Symmetric or not?
Step 1 - First find the transpose of the originally given matrix.
Step 2 – Then find the negative of the original matrix.
Step 3 – If the negative of the matrix obtained in Step2 is equal to the transpose of the matrix then the matrix is said to be skew-symmetric.
Properties:
Any matrix A can be written as a sum of /symmetric matrix and a skew-symmetric matrix.
\[A=\frac{1}{2}(A+A')+\frac{1}{2}(A-A')\]
Questions to solve
Question 1: Check whether the given matrices are symmetric or not.
M = \[\begin{bmatrix}0 & 5\\ 9 & 0\end{bmatrix}\]
P = \[\begin{bmatrix}1 & 4\\ 0 & -1\end{bmatrix}\]
Solution: We will first find the transpose of matrix M,
MT = \[\begin{bmatrix}0 & 9\\ 5 & 0\end{bmatrix}\]
Since the transpose of M is not equal to matrix M, therefore it is not a symmetric matrix.
We will first find the transpose of matrix P,
PT = \[\begin{bmatrix}1 & 0\\ 4 & -1\end{bmatrix}\]
Since the transpose of P is not equal to matrix P, therefore it is not a symmetric matrix.
Question 2 : Is the given matrix A, a skew-symmetric matrix. Give a reason for your answer.
\[A=\begin{bmatrix}0 & -1\\ 1 & -0\end{bmatrix}\]
Solution: First, we will find the transpose of the matrix A,
\[A^T=\begin{bmatrix}0 & 1\\ -1 & 0\end{bmatrix}\]
Now we will find the negative of the matrix A.
\[-A=\begin{bmatrix}0 & 1\\ -1 & 0\end{bmatrix}\]
Since, the negative of the matrix A is equal to the transpose of the matrix A.
Therefore, A is a skew-symmetric matrix.
Question 3: Show that the given matrix is a symmetric matrix.
\[A=\begin{bmatrix}1 & 2 & 3\\ 2 & 4 & 5\\ 3 & 5 & 8\end{bmatrix}\]
Solution: To check whether the given matrix A is a symmetric matrix,
We need to find the transpose of the given matrix A,
\[A^T=\begin{bmatrix}1 & 2 & 3\\ 2 & 4 & 5\\ 3 & 5 & 8\end{bmatrix}\]
Since the original matrix A is equal to the transpose matrix, therefore the given matrix A is a symmetric matrix.
Question 4: Check whether the given matrix B is a symmetric matrix or a skew-symmetric matrix.
\[B=\begin{bmatrix}0 & 5 & 3\\ -5 & 0 & -8\\ -3 & 8 & 0\end{bmatrix}\]
Solution: Let’s check whether the given matrix is symmetric or not.
We need to find the transpose of the given matrix B,
\[B^T=\begin{bmatrix}0 & -5 & -3\\ 5 & 0 & 8\\ 3 & -8 & 0\end{bmatrix}\]
Since the original matrix B is not equal to the transpose matrix (BT≠B), therefore the given matrix B is not a symmetric matrix.
Let’s check whether the given matrix is skew-symmetric or not.
Since we have already found the transpose,
\[B^T=\begin{bmatrix}0 & -5 & -3\\ 5 & 0 & 8\\ 3 & -8 & 0\end{bmatrix}\]
We will find the negative of the original matrix.
\[-B=\begin{bmatrix}0 & -5 & -3\\ 5 & 0 & 8\\ 3 & -8 & 0\end{bmatrix}\]
Since the negative of the matrix B is equal to the transpose of the matrix B.
Therefore, B is a skew-symmetric matrix.
Fun Facts about Matrices:
The term “Matrix” was coined by James Sylvester who was a 19th-century British Mathematician
The algebra of Matrices was developed by the mathematician Arthur Cayley who was James Sylvester’s friend.
Applications of Matrices:
Very often students studying Maths keep wondering about the practical applications of the concept. Thus, to do away with this wonderment, Vedantu has brought a list of practical and real-life applications of Matrices. These are as follows:
Matrices are used to represent real-world data like the population of people, maternal mortality rate, etc
Medical imaging, MRIs use matrices to operate
Matrices also have applications in creating video games
Electronics networks, aeroplanes and spacecraft, all require well-calibrated computations that are obtained from matrix transformations
Matrices are also used in the field of Economics
Matrix in computer graphics is used to convert geometric data into different coordinate systems.
Matrices are used in business as well. A decision matrix is used to make help the user make a complex decision
FAQs on Symmetric and Skew Symmetric Matrix
1. What are the defining algebraic conditions for a square matrix to be classified as symmetric or skew-symmetric?
A square matrix A is defined as symmetric if it is equal to its own transpose, satisfying the condition A = Aᵀ. This implies that for every element, aᵢⱼ = aⱼᵢ. Conversely, a square matrix A is defined as skew-symmetric if it is equal to the negative of its transpose, satisfying the condition A = -Aᵀ. This implies that for every element, aᵢⱼ = -aⱼᵢ.
2. How can any square matrix be uniquely expressed as the sum of a symmetric and a skew-symmetric matrix?
Any square matrix A can be uniquely represented as the sum of a symmetric matrix P and a skew-symmetric matrix Q. This is a fundamental theorem in matrix algebra, where the components are found using the formulas:
- Symmetric part: P = ½(A + Aᵀ)
- Skew-symmetric part: Q = ½(A - Aᵀ)
3. What is the mandatory property of the diagonal elements in any skew-symmetric matrix and why?
For any skew-symmetric matrix, all elements on the main diagonal must be zero. The reason stems from the defining condition aᵢⱼ = -aⱼᵢ. For diagonal elements, the row index equals the column index (i = j), leading to the equation aᵢᵢ = -aᵢᵢ. The only number that satisfies this equation is 0, as 2aᵢᵢ = 0 implies aᵢᵢ = 0.
4. What can be concluded about the determinant of a skew-symmetric matrix of odd order?
The determinant of any skew-symmetric matrix of an odd order 'n' is always zero. This is a critical property for competitive exams. The proof is as follows: We know that det(A) = det(Aᵀ). For a skew-symmetric matrix, Aᵀ = -A. Therefore, det(A) = det(-A). Using the determinant property det(kA) = kⁿdet(A), we get det(A) = (-1)ⁿdet(A). If 'n' is odd, this becomes det(A) = -det(A), which means 2det(A) = 0, proving that det(A) = 0.
5. If A and B are two symmetric matrices of the same order, under what specific condition is their product AB also symmetric?
The product AB is symmetric if and only if matrices A and B commute, which means AB = BA. To verify this, we check the transpose of the product: (AB)ᵀ = BᵀAᵀ. Since A and B are symmetric, Aᵀ = A and Bᵀ = B. Substituting these gives (AB)ᵀ = BA. For the product AB to be symmetric, we require (AB)ᵀ = AB. Therefore, the necessary and sufficient condition is AB = BA.
6. If a symmetric matrix A is invertible, is its inverse A⁻¹ also symmetric?
Yes, if an invertible matrix A is symmetric, its inverse A⁻¹ is also guaranteed to be symmetric. This can be proven using the property of transposes and inverses: (A⁻¹)ᵀ = (Aᵀ)⁻¹. Since A is symmetric, we know that Aᵀ = A. By substituting this into the equation, we get (A⁻¹)ᵀ = A⁻¹, which is the definition of a symmetric matrix.
7. If A is a skew-symmetric matrix, what can be determined about the nature of its powers, such as A² and A³?
The powers of a skew-symmetric matrix A follow a consistent pattern regarding their symmetry:
- A² is a symmetric matrix. Proof: (A²)ᵀ = (Aᵀ)² = (-A)² = A².
- A³ is a skew-symmetric matrix. Proof: (A³)ᵀ = (Aᵀ)³ = (-A)³ = -A³.
8. What is the significance of the fact that every real symmetric matrix is diagonalizable?
The fact that every real symmetric matrix is orthogonally diagonalizable is a result of the Spectral Theorem. Its significance for JEE Advanced is profound: it guarantees that for any symmetric matrix, a set of orthogonal eigenvectors can be found. This property is essential for simplifying complex systems, finding principal axes in rigid body dynamics, and solving systems of linear differential equations that appear in physics and engineering problems.
9. How does the trace of a square matrix relate to the trace of its symmetric and skew-symmetric parts?
The trace of a matrix is fully contained within its symmetric part, while the trace of its skew-symmetric part is always zero. Let A = P + Q, where P is symmetric and Q is skew-symmetric.
- Tr(P) = Tr(A), because Tr(P) = Tr(½(A + Aᵀ)) = ½(Tr(A) + Tr(Aᵀ)) = Tr(A).
- Tr(Q) = 0, because the diagonal elements of a skew-symmetric matrix are all zero.
