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Matrix is an important aspect of Mathematics and any stream of study remotely related to Mathematics. A matrix having m rows and n columns is called a matrix of order m × n or simply m × n matrix. A square matrix is a matrix with the same number of rows and columns. An n-by-n matrix is known as a square matrix of order n. Any two square matrices of the same order can be added and multiplied.
Hence, before getting into the main topic, Symmetric and Skew Symmetric Matrices, students must make a note of the following points:
- A matrix having m rows and n columns is called a matrix of order m × n or simply m × n matrix.
- A square matrix is a matrix with the same number of rows and columns. An n-by-n matrix is known as a square matrix of order n.
- Any two square matrices of the same order can be added and multiplied.
- The transpose of a matrix is a new matrix whose rows are the columns of the original. (This makes the columns of the new matrix the rows of the original).
- According to matrices, only square matrices can be symmetric or skew-symmetric in form.
| Table of Content |
Key Terms: Matrix, Symmetric Matrix, Skew Symmetric Matrix, Transpose, Properties, Determinant
Read More: Matrices Handwritten Notes
Transpose of a Matrix
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The transpose of a matrix is a new matrix whose rows are the columns of the original.
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This makes the columns of the new matrix the rows of the original.
The superscript "T" or " ' " represents the transpose of a matrix.
Transpose of a Matrix
The video below explains this:
Matrices Detailed Video Explanation:
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Matrix Multiplication
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- The number of columns in the first matrix must be equal to the number of rows in the second matrix.
- The resulting matrix has the number of rows of the first and the number of columns of the second matrix.
Matrix Multiplication
Symmetric Matrix
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For a matrix B, if B = B’ (Matrix B = Transpose of Matrix B), that is whenever the transpose of a matrix is equal to it, the matrix is known as a symmetric matrix.
Symmetric Matrix
The entries of a symmetric matrix are symmetric with respect to the main diagonal.
Properties of Symmetric Matrix
- The addition and difference of two symmetric matrices result in a symmetric matrix.
- If A and B are two symmetric matrices and they follow the commutative property, i.e. AB = BA, then the product of A and B is symmetric.
- If matrix A is symmetric then An is also symmetric, where n is an integer.
- If A is a symmetric matrix then A-1 is also symmetric.
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Skew-Symmetric Matrix
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A skew-symmetric matrix (also known as antisymmetric or antimetric) is a square matrix whose transpose equals the negative of the matrix.
The diagonal elements of a skew-symmetric matrix are equal to zero.
Properties of a skew-symmetric matrix
- When we add two skew-symmetric matrices then the resultant matrix is also skew-symmetric.
- The scalar product of the skew-symmetric matrix is also a skew-symmetric matrix.
- The diagonal of the skew-symmetric matrix consists of zero elements and therefore the sum of elements in the main diagonals is equal to zero.
- When the identity matrix is added to the skew-symmetric matrix then the resultant matrix is invertible.
- The determinant of the skew-symmetric matrix is non-negative.
Determinant of Skew Symmetric Matrix
Let A be an n cross n skew-symmetric matrix. The determinant of A satisfies
All odd dimension skew-symmetric matrices are singular as their determinants are always zero.
Things to Remember
- The transpose of a matrix is a new matrix whose rows are the columns of the original.
- A square matrix is a matrix with the same number of rows and columns.
- When we add two skew-symmetric matrices then the resultant matrix is also skew-symmetric.
- The scalar product of the skew-symmetric matrix is also a skew-symmetric matrix.
- Any two square matrices of the same order can be added and multiplied.
- All odd dimension skew-symmetric matrices are singular as their determinants are always zero.
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| Upper Triangular Matrix | Orthogonal Matrix | Matrix Multiplication |
| Symmetric and Skew Symmetric Matrices | Determine the Order of Matrix | Determinants and Matrices |
Sample Questions
Ques. What are the best study materials for this topic? (1 Mark)
Ans. Students' first priority must be solving the NCERT textbook-based questions.
Ques. Is there any relation between the Symmetric and Skew-symmetric matrix? (1 Mark)
Ans. The relation between them is that "Every square matrix can be expressed uniquely as the sum of symmetric and skew-symmetric matrices".
Ques. What are the types of matrices? (4 Marks)
Ans. The major types of matrices are:
- Row Matrix: A matrix that contains only one row and any no of columns is known as a row matrix.
- Column Matrix: A matrix that contains only one column and any no of rows is known as a column matrix.
- Singleton Matrix: A matrix that has only one element is known as a singleton matrix.
- Rectangular Matrix: A matrix that does not have an equal number of rows and columns.
- Square Matrix: A matrix that has an equal number of rows and an equal number of columns
Ques. Can a matrix be both symmetric and skew-symmetric? (1 Mark)
Ans. Yes. Zero or Null matrix are the only matrices that are both Symmetric and Skew-symmetric.
Ques. Suggest some good books for NCERT Maths? (2 Marks)
Ans. The students can refer to these popular books:
- RS Aggarwal
- RD Sharma
The students are also advised to solve previous year papers along with the sample papers in order to gain confidence and get an idea about the type of questions asked.
Ques. If the matrix A = \(\begin{bmatrix} 0 & a & -3 \\ 2 & 0 & -1 \\ b & 1 & 0 \end{bmatrix}\) is skew-symmetric, find the value of ‘a’ and ‘b’. (2018) (3 Marks)
Ans. Given, A = \(\begin{bmatrix} 0 & a & -3 \\ 2 & 0 & -1 \\ b & 1 & 0 \end{bmatrix}\)
A is given to be skew-symmetric matrix.
∴ AT = –A
∴ \(\begin{bmatrix} 0 & 2 & b \\ a & 0 & 1 \\ -3 & -1 & 0 \end{bmatrix}\) = –\(\begin{bmatrix} 0 & a & -3 \\ 2 & 0 & -1 \\ b & 1 & 0 \end{bmatrix}\)
⇒ \(\begin{bmatrix} 0 & 2 & b \\ a & 0 & 1 \\ -3 & -1 & 0 \end{bmatrix}\) = \(\begin{bmatrix} 0 & -a & 3 \\ -2 & 0 & 1 \\ -b & -1 & 0 \end{bmatrix}\)
On comparing both sides we get,
-a = 2 and -b = -3
⇒ a = -2 and b = 3
Ques. If A is a skew-symmetric matrix of order 3, then prove that det A = 0. (2017 outside Delhi) (3 Marks)
Ans. We know A is a skew-symmetric matrix of 3rd order
AT = -A
Now |AT| = |-A|
|AT| = (-1)³ |A| [1|kA| = kn |A| where n can be said as the order of A]
|A| = -|A| [IAT| = |A|]
|A| + |A| = 0
Therefore 2|A| = 0 or |A| = 0
i.e., det A = 0
Hence proved.
Ques. Matrix A = \(\begin{bmatrix} 0 & 2b & -2 \\ 3 & 1 & 3 \\ 3a & 3 & -1 \end{bmatrix}\) is given to be symmetric, find values of a and b. (3 Marks)
Ans. We have, A = \(\begin{bmatrix} 0 & 2b & -2 \\ 3 & 1 & 3 \\ 3a & 3 & -1 \end{bmatrix}\)
It is given that the matrix is symmetric.
∴ A = A′
⇒ \(\begin{bmatrix} 0 & 2b & -2 \\ 3 & 1 & 3 \\ 3a & 3 & -1 \end{bmatrix}\) = \(\begin{bmatrix} 0 & 3 & 3a \\ 2b & 1 & 3 \\ -2 & 3 & -1 \end{bmatrix}\)
Now, by equality of matrices, we get
2b = 3
⇒ b = \(\frac{3}{2}\)
and 3a = -2
⇒ a = \(\frac{-2}{3}\)
Therefore, a = \(\frac{-2}{3}\) and b = \(\frac{3}{2}\)
Ques. If A = \(\begin{bmatrix} 3 & 5 \\ 7 & 9 \end{bmatrix}\) is written as A = P + Q, where P is a symmetric matrix and Q is a skew-symmetric matrix, write the matrix P. (2016 foreign) (3 Marks)
Ans. Given,
A = \(\begin{bmatrix} 3 & 5 \\ 7 & 9 \end{bmatrix}\)
P is symmetric matrix. So, P = \(\frac{1}{2}(A+A^T)\)
Q is skew symmetric matrix. So, Q = \(\frac{1}{2}(A-A^T)\)
AT = \(\begin{bmatrix} 3 & 7 \\ 5 & 9 \end{bmatrix}\)
P = \(\displaystyle \frac{1}{2}\begin{bmatrix} 6&12\\ 12&18 \end{bmatrix}\) = \(\begin{bmatrix} 3 & 6 \\ 6 & 9 \end{bmatrix}\)
Ques. Write a 2×2 matrix which is both symmetric and skew-symmetric. (2014 Delhi) (1 Mark)
Ans.The matrix which is both symmetric as well as skew-symmetric at the same time is referred to as a null matrix. Hence the matrix looks something like this:
\(\begin{bmatrix} 0&0 \\ 0&0 \end{bmatrix}\)
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