If A and B are square matrices of order 3 such that|A| = –1, |B| = 3, then |3 AB| is equal to

If A and B are square matrices of order 3 such that
|A| = –1, |B| = 3, then |3 AB| is equal to
(a) –9 (b) –81 (c) –27 (d)81

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Matrices
A matrix is an ordered rectangular array of numbers or functions. The numbers or functions are called the elements or the entries of the matrix. We denote matrices by capital letters.
the horizontal lines of elements are said to constitute, rows of the matrix and the vertical lines of elements are said to constitute, columns of the matrix.
Order of a matrix
A matrix having m rows and n columns is called a matrix of order m × n or simply m × n matrix (read as an m by n matrix).
1. We shall follow the notation, namely A = [aij]m × n to indicate that A is a matrix of order m × n.
2. We shall consider only those matrices whose elements are real numbers or functions taking real values.
Types of Matrices
(i) Column matrix
(ii) Row matrix
(iii) Square matrix
(iv) Diagonal matrix
(v) Scalar matrix
(v) Scalar matrix
(vi) Identity matrix
(vii) Zero matrix

Equality of matrices
Two matrices A = [aij] and B = [bij] are said to be equal if
(i) they are of the same order
(ii) each element of A is equal to the corresponding element of B, that is aij = bij for all i and j.

Operations on Matrices
Multiplication of a matrix by a scalar
multiplication of a matrix by a scalar
In other words, kA = k [aij] m × n = [k (aij)] m × n, that is, (i, j)th element of kA is kaij
for all possible values of i and j.

Difference of matrices If A = [aij], B = [bij] are two matrices of the same order,
say m × n, then difference A – B is defined as a matrix D = [dij], where dij = aij – bij,
for all value of i and j. In other words, D = A – B = A + (–1) B, that is sum of the matrix
A and the matrix – B.
Properties of matrix addition
(i) Commutative Law If A = [aij], B = [bij] are matrices of the same order, say
m × n, then A + B = B + A.
ii) Associative Law For any three matrices A = [aij], B = [bij], C = [cij] of the
same order, say m × n, (A + B) + C = A + (B + C).
(iii) Existence of additive identity Let A = [aij] be an m × n matrix and
O be an m × n zero matrix, then A + O = O + A = A
(iv) The existence of additive inverse Let A = [aij]m × n be any matrix, then we
have another matrix as – A = [– aij]m × n such that A + (– A) = (– A) + A= O. So
– A is the additive inverse of A or negative of A.
Properties of scalar multiplication of a matrix
If A = [aij] and B = [bij] be two matrices of the same order, say m × n, and k and l are
scalars, then
(i) k(A +B) = k A + kB, (ii) (k + l)A = k A + l A
(ii) k (A + B) = k ([aij] + [bij])
= k [aij + bij] = [k (aij + bij)] = [(k aij) + (k bij)]
= [k aij] + [k bij] = k [aij] + k [bij] = kA + kB
(iii) ( k + l) A = (k + l) [aij]
= [(k + l) aij] + [k aij] + [l aij] = k [aij] + l [aij] = k A + l A

Multiplication of matrices
Properties of multiplication of matrices
1. The associative law (AB) C = A (BC)
2. The distributive law
(i) A (B+C) = AB + AC
(ii) (A+B) C = AC + BC
3. The existence of multiplicative IA = AI = A.
Transpose of a Matrix
If A = [aij] be an m × n matrix, then the matrix obtained by interchanging
the rows and columns of A is called the transpose of A. Transpose of the matrix A is
denoted by A′or (AT). In other words, if A = [aij]m × n, then A′= [aji]n × m.
Properties of transpose of the matrices
(i) (A’)′ = A, (ii) (kA)′ = kA′ (where k is any constant)
(iii) (A + B)′ = A′ + B′ (iv) (A B)′ = B′ A′

Symmetric and Skew Symmetric Matrices
-A square matrix A = [aij] is said to be symmetric if A′ = A, that is,
[aij] = [aji] for all possible values of i and j.
-A square matrix A = [aij] is said to be skew symmetric matrix if
A′ = – A, that is aji = – aij for all possible values of i and j. Now, if we put i = j, we
have aii = – aii. Therefore 2aii = 0 or aii = 0 for all i’s.

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

Elementary Operation (Transformation) of a Matrix
(i) The interchange of any two rows or two columns
(ii) The multiplication of the elements of any row or column by a non zero number.
The corresponding column operation is denoted by Ci → kCi
(iii) The addition to the elements of any row or column, the corresponding elements of any other row or column multiplied by any non zero number.

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