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Matrices Class 12 Notes Mathematics
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CSBE Class 12 Mathematics Chapter 3 Matrices
- A matrix is an ordered rectangular array of numbers, real or complex or functions.
- A matrix having m rows and n columns is called a matrix of order m × n.
- Column matrix: A matrix with one column is denoted by .
- Row matrix: A matrix with one row is denoted by .
- Square matrix: An m × n matrix is a square matrix if m = n.
- Diagonal matrix: A = is a diagonal matrix if =0, when
- Scalar matrix: is a scalar matrix if when i ≠ j, (k is some constant), when I=j.
- Identity matrix: is an identity matrix, if
- Zero matrix: A zero matrix has all its elements as zero.
- Equality of two matrices: if (i) A and B are of same order, (ii) for all possible values of
- Scalar multiplication:
Also – A = (–1)A
- A – B = A + (–1) B
A + B = B + A
(A + B) + C = A + (B + C), where A = B = and C = are of same order.
- k(A + B) = kA + kB, where A and B are of same order, k is constant.
- (k + l ) A = kA + lA, where k and l are constant.
- (i) A(BC) = (AB)C,
(ii) A(B + C) = AB + AC,
(iii) (A + B)C = AC + BC
- (i) (A′)′ = A, · (ii) (kA)′ = kA′, · (iii) (A + B)′ = A′ + B′, · (iv) (AB)′ = B′A′
- Symmetric matrix: A is a symmetric matrix if A′ = A.
- Skew-aymmetric matrix: A is a skew symmetric matrix if A′ = –A.
- Any square matrix can be represented as the sum of a symmetric and a skew symmetric matrix. In fact, A = (A + A’) + (A – A’), where (A + A’) is a symmetric matrix and (A – A’) is a skew-symmetric matrix.
- Equivalent matrices: Two matrices A and B are equivalent that is, A – B is A is obtained from the other by a sequence of elementary operations. Elementary operations of a matrix are as follows:
(i) or (interchange rows or columns)
- If A and B are two square matrices such that AB = BA = I, then B is the inverse matrix of A and is denoted by and A is the inverse of B.
- Inverse of a square matrix, if it exists, is unique.
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