Summary Mathematics/2023 NCERT part 1 -:matrices and determinants

MATRICES
A rectangular arrangement of numbers (elements) in rows and columns is called a Matrix.

Order of A matrix with m rows and n columns is called as matrix with order m X n
Matrix A = [aij]m X n

Construction of Matrix
Construct a matrix of order m X n with A = [a ij] where i represents the respective row and j represents the
respective column

Types of Matrices
Row Matrix A = [aij]m X n with m =1

Column Matrix A = [aij]m X n with n =1

Singleton Matrix A = [aij]m X n with m = n = 1

Null Matrix A = [aij]m X n, aij = 0 for all i and j

Square Matrix A = [aij]m X n with m = n

In a square matrix if all the elements other the principal diagonal elements are 0
i.e. in A = [aij]m X n, aij = 0 for all i ≠ j

 The elements of a square Matrix for which i = j are called as diagonal elements. The line
joining these elements is called the principal diagonal.
 No element of principal diagonal in diagonal matrix is zero
 No. of zero’s = n2-n

Diagonal Matrix Scalar If all the elements of principal diagonal in diagonal matrix are equal
Matrix i.e. in A = [aij]m X m, aij = 0 for all i ≠ j
= K for all i = j

If all the elements of principal diagonal are 1 and all the other elements
Unit/Identity are 0.
Matrix i.e. in A = [aij]m X m, aij = 0 for all i ≠ j
= 1 for all i = j

Upper triangular A square matrix [aij] with aij = 0 when i > j
Matrix

Lower triangular A square matrix [aij] with aij = 0 when i < j
Matrix

Equal Matrix Two matrices are equal if they are of same order and their corresponding elements are

, equal.

Periodic matrices Ak+1 = A, K ≥ 1; where k is the period of the matrix

Nilpotent matrices An = 0, An+1 = 0, An+2 = 0 but An-1 ≠ 0 (n -> Index of matrix)
 IAI=0

Orthogonal matrices are square matrices which, when multiplied with its transpose matrix
results in an identity matrix.
i.e. A. AT = I = AT. A
Orthogonal Or AT = A-1
matrices [a1 a2 a3]
A = [b1 b2 b3] then a12 + a22 + a32 = b12 + b22 + b32 = c12 + c22 + c32 = 1 and
[c1 c2 c3] a1b1 + a2b2 + a3b3 = b1c1 + b2c2 + b3c3 = a1c1 + a2c2 + a3c3 = 0
 IAI=±1

Idempotent matrices are square matrices which, when multiplied with itself results in the
given matrix.
Idempotent i.e. A2 = A
2 3 4 5
Matrices  A=A =A =A =A …
 I A I = 0 or 1
 Idempotent matrix is either singular or Identity matrix.

Involuntary matrices are square matrices in which
Involuntary A2 = I Or A-1 = A
Matrices  Aodd = A; Aeven = I
 IAI=±1

ADDITION AND SUBTRACTION OF MATRICES
If A = [aij] m x n and B = [bij] m x n are two matrices with same order then
A + B= [aij + bij] m x n
A - B= [aij - bij] m x n

Properties of Matrix Addition
Commutative Property A+B=B+A
Associative Property A + (B + C) = (A + B) + C
Additive identity A + O = A ( where O is an additive identity)
Additive inverse Additive inverse of A will be -A
Cancellation Law A + B = A + C => B = C
B + A = C + A => B = C

Addition and Subtraction of diagonal Matrices
If A = Diag (a1, a2, a3….an); B = Diag (b1, b2, b3….bn) then
A + B = Diag (a1+ b1, a2+ b2, a3+ b3….an+ bn)
A - B = Diag (a1- b1, a2- b2, a3- b3….an- bn)

, SCALAR MULTIPLICATION OF MATRICES
If A = [aij]m × n is a matrix and k is a scalar, then kA = k [a ij]m×n

Properties of Scalar Multiplication of a Matrix
k(A + B) = kA + kB, (k + l) A = kA + lA (K.I) A = K.(I.A) = I. (K.A) (-K). A = (-K. A) = K.(-A) (-1). A = -A

MULTIPLICATION OF MATRICES
If A = [aij] m X n & B = [bij] p X q then A X B will be defined only when n = p. A X B will be of order m X q

Properties of Multiplication of a Matrix
Not Commutative AB ≠ BA
Associative A(BC) = (AB)C [Order remain same]
Distributive  A (B + C) = AB + AC (Multiplication from left)
 (A + B) C = AC + BC (Multiplication from right)
Multiplicative Im A = A In = A
No Cancellation law AB = BC => A ≠ C
AB = 0 Does not mean either A = 0 or B = 0
Multiplication of two diagonal matrices always result in diagonal matrices
Multiplication of two triangular matrices always result in triangular matrices
Multiplication of two scalar matrices always result in scalar matrices

If A & B are of same order then
If A & B are commutative (AB = BA) If A & B are anti commutative (AB = -BA)
(A+B)2 = A2 + B2 + AB + BA = A2 + B2 + 2AB (A+B)2 = A2 + B2 + AB + BA = A2 + B2
(A-B)2 = A2 + B2 - AB – BA = A2 + B2 - 2AB (A-B)2 = A2 + B2 - AB – BA = A2 + B2
(A-B) (A+B) = A2 - B2 + AB – BA = A2 - B2 (A-B) (A+B) = A2 - B2 + AB – BA = A2 - B2 + 2AB
2 2 2 2
(A+B) (A-B) = A - B - AB + BA = A - B (A+B) (A-B) = A2 - B2 - AB + BA = A2 - B2 + 2BA
2 2
(I + A) = I + 2A + A
For Practice
1. If 3A – B = [5 0] and B = [4 3] Then write the order of matrix A.
[1 1] [2 5]
Then find the value of matrix A. 6. Write the element a of a 3 × 3 matrix A = [a ij],
whose elements are given by aij = |i−j|/2
2. Find the value of x – y, if
2 [1 3] + [y 0] = [5 6] 7. If [2x 3] [1 2] [x] = 0, find x.
[0 x] [1 2] [1 8] [-3 0] [3]

3. If A is a square matrix such that A 2 = I, then find 8. Find the value of x – y, if
the simplified value of (A – I)3 + (A + I)3 – 7A. 2 [3 4] + [1 y] = [7 0]
[5 x] [0 1] [10 5]
4. Write the number of all possible matrices of order
2 × 2 with each entry 1, 2 or 3. 9. Solve the following matrix equation for x.
[x 1] [1 0] = 0
5. If [2 1 3] [-1 0 -1] [1] = A [-2 0]
[-1 1 0] [0]
[0 1 1] [1]