MATRICES A rectangular arrangement of elements is called matrix. Types of matrices: Null matrix: A matrix whose all elements are zero is called a null matrix or zero matrix. Upper triangular matrix: In an upper triangular matrix, all elements below the main diagonal are zero. Lower triangular matrix: In a lower triangular matrix, all elements above the main diagonal are zero.

ALGEBRA OF MATRICES PROPERTIES OF MATRIX ADDITION (i) matrix addition is commutative i.e. if A and B are two m x n matrices then A + B = B + A (ii) matrix addition is associative i.e. if A, B and C are three matrices of the same order then (A + B) + C = A + (B +C). (iii) Existence of identity The null matrices is the identity element for matrix addition i.e A + O = A = O +A. (IV) Existence of Inverse For every matrix A there exit a matrix –A such that A+(-A) = O. (v) Cancellation laws hold good in case of addition of matrices If A, B and C are three matrices of the same order then A + B = A + C =» B =C (left cancellation law)

B + A =C + A =» B =C (right cancellation law) MULTIPLICATION OF MATRICES Remark: If A and B are two matrices such that AB exists, then BA may or may not exist. TRANSPOSE OF A MARTIX A T is obtained from A by changing its rows into columns and its columns into rows. PROPERTIES OF TRANSPOSE Let A and B be two matrices. Then (i) (A T ) T = A (ii) (A+B) T = A T + B T, A and B being of same order. (iii) (kA) T = k A T, k be any scaler (iv) (AB) T = B T A T

Symmetric Matrix: A square matrix A is a symmetric matrix iff A T = A. Skew-symmetric Matrix: A square matrix A is a skew-symmetric matrix iff A T = -A. DETERMINANT Every square matrix can be associated to an expression or a no. known as determinant. Denoted by |A| Singular matrix: A square matrix is a singular matrix if its determinant is zero. Otherwise, it is a non-singular matrix. PROPERTIES OF DETERMINANTS P-1 Let A = [a ij ] be a square matrix of order n, then the sum of the product of elements of any row (column) with their cofactors is always equal to |A|.

P-2 Let A = [a ij ] be a square matrix of order n, then the sum of the products of elements of any row (column) with the cofactors of the corresponding elements of some other row (column) is zero. P-3 Let A = [a ij ] be a square matrix of order n, then |A| = |A T |. P-4 Let A = [a ij ] be a square matrix of order n(≥2) and let B be a matrix obtained from A by interchanging any two rows (columns) of A, then |B| = - |A|. P-5 If any two rows (columns) of a square matrix A = [a ij ] be a square matrix of order n(≥2) are identical, then its determinant is zero i.e. |A| = 0. P-6 Let A = [a ij ] be a square matrix of order n, and B be the matrix obtained from A by multiplying each element of a row (column) of A by a scalar k, then |B| = k |A|.

Inverse of a matrix A square matrix of order n is invertible if there exists a square matrix B of the same order such that AB = I n = BA. In such a case, we say that the inverse of A is B and we write, A -1 = B. Algorithm for finding the inverse of a square matrix A STEP 1 Find |A| STEP 2 If |A| = 0, then write” A is a singular matrix and hence not invertible” STOP Else write “A is a non-singular & hence invertible ”. STEP 3 Calculate the cofactors of elements of A STEP 4 Write the matrix of cofactors of elements of A and then obtain its transpose to obtain adj A. STEP 5 Find the inverse of A by using the formula A -1 = 1 adj A |A| STOP