NCERT Class 12-Mathematics: Exemplar Chapter – 3 Matrices Part 2 (For CBSE, ICSE, IAS, NET, NRA 2023)
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3.1. 9 Symmetric Matrix and Skew Symmetric Matrix
(i) A square matrix is said to be symmetric if , that is, for all possible values of and .
(ii) A square matrix is said to be skew symmetric matrix if , that is for all possible values of and .
Note: Diagonal elements of a skew symmetric matrix are zero.
(iii) Theorem 1: For any square matrix A with real number entries, is a symmetric matrix and is a skew symmetric matrix.
(iv) Theorem 2: Any square matrix A can be expressed as the sum of a symmetric matrix and a skew symmetric matrix, that is
3.1. 10 Invertible Matrices
(i) If A is a square matrix of order , and if there exists another square matrix B of the same order , such that , then, A is said to be invertible matrix and B is called the inverse matrix of A and it is denoted by .
Note :
1. A rectangular matrix does not possess its inverse, since for the products BA and AB to be defined and to be equal, it is necessary that matrices A and B should be square matrices of the same order.
2. If B is the inverse of A, then A is also the inverse of B.
(ii) Theorem 3: (Uniqueness of inverse) Inverse of a square matrix, if it exists, is unique.
(iii) Theorem 4: If A and B are invertible matrices of same order, then
3.1. 11 Inverse of a Matrix using Elementary Row or Column Operations
To find using elementary row operations, write and apply a sequence of row operations on till we get, . The matrix B will be the inverse of A. Similarly, if we wish to find using column operations, then, write and apply a sequence of column operations on till we get, .
Note: In case, after applying one or more elementary row (or column) operations on , if we obtain all zeros in one or more rows of the matrix A on L. H. S. , then does not exist.
3.2 Solved Examples
Short Answer (S. A)
Question 1:
Construct a matrix whose elements are given
Answer:
Question 2:
If then which of the sums is defined?
Answer:
Only is defined since matrices of the same order can only be added.
Question 3:
Show that a matrix which is both symmetric and skew symmetric is a zero matrix.
Answer:
Let be a matrix which is both symmetric and skew symmetric.
Since A is a skew symmetric matrix, so .
Thus for all and , we have
Again, since A is a symmetric matrix, so .
Thus, for all and , we have
Therefore, from and , we get
for all and
i.e.. , for all i and j. Hence A is a zero matrix.
Question 4:
If , Find the value of
Answer:
We have
Question 5:
If A is invertible matrix, then show that for any scalar (non-zero) , is invertible and
Answer:
We have
Hence (kA) is inverse of