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Symmetric and Skew-Symmetric Matrices
As we are all aware by now that equal matrices have equal dimensions, hence only the square matrices can be symmetric or skew-symmetric form. Now, what is a symmetric matrix and a skew-symmetric matrix? Let’s learn about them in further detail below.

Symmetric Matrix

A square matrix A is said to be symmetric if aij = aji for all i and j, where aij is an element present at (i,j)th position (ith row and jth column in matrix A) and aji is an element present at (j,i)th position (jth row and ith  column in matrix A). In other words, we can say that matrix A is said to be symmetric if the transpose of matrix A is equal to matrix A itself (AT=A). Let’s take an example of a matrix,

Matrices Part 2 | Algebra - Mathematics

It is symmetric matrix because aij = aji for all i and j. Here, a12 = a21= 3, a13 = a31= 8 and a23 = a32 = -4 In other words, the transpose of Matrix A is equal to Matrix A itself (AT=A) which means matrix A is symmetric.

Skew-Symmetric Matrix
Square matrix A is said to be skew-symmetric if aij =−aji for all i and j. In other words, we can say that matrix A is said to be skew-symmetric if transpose of matrix A is equal to negative of matrix A i.e (AT =−A). Note that all the main diagonal elements in the skew-symmetric matrix are zero. Let’s take an example of a matrix

Matrices Part 2 | Algebra - Mathematics

It is skew-symmetric matrix because aij =−aji for all i and j. Here, a12 = -6 and a21= -6 which means  a12= −a21. Similarly, this condition holds true for all other values of i and j.

Theorem 1
For any square matrix A with real number entries, A + A′ is a symmetric matrix and A – A′ is a skew-symmetric matrix.

Proof:  Let B =A+A′, then B′= (A+A′)′
=  A′ + (A′)′ (as (A + B)′ = A′ + B′)
=  A′ +A (as (A′)′ =A)
=  A + A′ (as A + B = B + A) =B
Therefore, B = A+A′is a symmetric matrix

Now let C = A – A′
C’ = (A–A′)′=A′–(A′)′ (Why?)
= A′ – A (Why?)
=– (A – A′) = – C
Hence, A – A′ is a skew-symmetric matrix.

Theorem 2
Any Square matrix can be expressed as the sum of a symmetric and a skew-symmetric matrix.

Proof: Let A be a square matrix then, we can write A = 1/2 (A + A′) + 1/2 (A − A′). From the Theorem 1, we know that (A + A′) is a symmetric matrix and (A – A′) is a skew-symmetric matrix.

Since for any matrix A, (kA)′ = kA′, it follows that 1 / 2 (A+A′) is a symmetric matrix and 1 / 2 (A − A′) is a skew-symmetric matrix. Thus, any square matrix can be expressed as the sum of a symmetric and a skew-symmetric matrix.

Solved Examples for You
Question: If A and B are symmetric matrices, then ABA is

  1. Symmetric
  2. Skew – Symmetric
  3. Diagonal
  4. Triangular

Solution: Given A and B are Symmetric Matrices
⇒ AT = A and BT = B
Now, take (ABA)T
(ABA)T = ATBTAT
(ABA)T = ABA
Hence, ABA is also Symmetric

Question: Say true or false: If A & B are symmetric matrices of same order then AB − BA is symmetric.

  1. True
  2. False

Solution: Given, A and B are symmetric matrices, therefore we have:
A’ = A and B’ = B……….(i)
Consider, (AB – BA)’ = (AB)’ – (BA)’……………[ Since, (A – B)’ = A’ – B’]
= B’A’ – A’ B’ ……………[ Since, (AB)’ = B’ A’]
= BA – AB …..[by (i)]
= – (AB – BA)
Therefore, (AB – BA)’ = – (AB – BA)
Thus, (AB – BA) is a skew-symmetric matrix.

Elementary Operation of a Matrix
Elementary Operations! Let’s get a deeper understanding of what they actually are and how are they useful. Elementary operations for matrices play a crucial role in finding the inverse or solving linear systems. They may also be used for other calculations. The matrix on which elementary operations can be performed is known as an elementary matrix.

What is an Elementary Matrix?
An elementary matrix is a matrix which differs from the identity matrix by one single elementary row operation (or column operation). Now, let’s consider a matrix given below,

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FAQs on Matrices Part 2 - Algebra - Mathematics

1. What is the purpose of matrices in mathematics?
Matrices are used in mathematics to organize and manipulate data efficiently. They provide a way to represent and solve systems of linear equations, perform transformations in geometry, and solve complex problems in various fields such as physics, economics, and computer science.
2. How do you multiply matrices?
To multiply matrices, you need to ensure that the number of columns in the first matrix is equal to the number of rows in the second matrix. Each element in the resulting matrix is obtained by multiplying the corresponding elements in the rows of the first matrix with the columns of the second matrix and summing them up.
3. Can matrices be added or subtracted?
Yes, matrices can be added or subtracted if they have the same dimensions. The addition or subtraction is done by simply adding or subtracting the corresponding elements in the matrices. The resulting matrix will have the same dimensions as the original matrices.
4. What is the determinant of a matrix?
The determinant of a matrix is a scalar value that can be calculated for square matrices. It provides important information about the matrix, such as whether it is invertible or singular. The determinant is calculated using a specific formula that involves the elements of the matrix.
5. How can matrices be used in solving systems of linear equations?
Matrices provide an efficient way to solve systems of linear equations. The system of equations can be represented as a matrix equation, where the coefficients of the variables form a matrix. By performing row operations on the augmented matrix, the system can be transformed into row-echelon form or reduced row-echelon form, allowing for the solution to be easily determined.
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