By going through these CBSE Class 12 Maths Notes Chapter 3 Matrices, students can recall all the concepts quickly.

## Matrices Notes Class 12 Maths Chapter 3

**Matrix (Definition):** A matrix is defined as a rectangular array (arrangement) of numbers or functions.A

The matrices are denoted by capital letters as shown below:

→ Elements: The numbers or functions in a matrix are called its elements. In matrix A; 2, 5, 6, 4, 0 and \(\sqrt{3}\) are the elements.

→ Row: The elements lying in a horizontal line form a row. Matrix B has 3 rows viz: first row is (1, 3 + 2i, \(\frac{2}{3}\)) is (-5, 2.3, 7) and third row is (\(\sqrt{7}\) 4 -8).

→ Column: The elements lying in a vertical line form a column. Matrix A has three columns viz: first column is \(\left(\begin{array}{l}

2 \\

4

\end{array}\right)\), second is \(\left(\begin{array}{l}

5 \\

0

\end{array}\right)\) and third is \(\left(\begin{array}{c}

6 \\

\sqrt{3}

\end{array}\right)\)

→ Order of matrix: A matrix, having m rows and n columns, is said to be of the order m × n. The matrix A is of order 2 × 3, B is of order 3 × 3 and C is of order 3 × 2.

In general, a matrix of order m × n, i.e., consisting of m rows and n columns is denoted by A = [a_{ij}]_{m×n}.

A number of elements in the matrix [a_{ij}]_{m×n} are m × n and nth element = a_{ij} is that element that lies in the ith row and jth column.

**Types of matrices:**

→ Square Matrix: If in a matrix, the number of rows is equal to the number of columns, then the matrix is called a square matrix.

has 3 rows and 3 columns. Therefore, it is a square matrix.

In general, [a_{ij}]_{n×n} is a square matrix of order n. the elements a_{11}, a_{22}, a_{33},…,a_{ii}…, a_{nn} are the elements of main diagonal. Thus, in the matrix P; 2, 7 and 1 are the diagonal elements.

→ Row Matrix: A matrix, which has one row is known as row matrix. [3 -1 i 2] is a row matrix, which has only one row.

→ Column Matrix: A matrix having one column is said to be is a column matrix.\(\left[\begin{array}{c}

-1 \\

3 \\

2

\end{array}\right]\) is a column matrix, since there is only one column in it.

→ Diagonal Matrix: A square matrix is called a diagonal matrix, if its non-diagonal elements are zero, i.e., a_{ij} = 0, when i ≠ 0, e.g. \(\left[\begin{array}{ll}

2 & 0 \\

0 & 1

\end{array}\right]\) is a diagonal matrix.

→ Scalar Matrix: It is a square matrix whose (a) diagonal elements are non-zero and equal (b) non-diagonals elements are zero, Le, a_{ij} = k ≠ 0 when j = j, a_{ij} = 0, when i ≠ j.\(\left[\begin{array}{lll}

2 & 0 & 0 \\

0 & 2 & 0 \\

0 & 0 & 2

\end{array}\right]\) is scalar matrix.

→ Unit or Identity Matrix: It is a square matrix in which each diagonal element is 1. i.e., a_{ij} = 1 when i = j and aij = 0 when i ≠ j.\(\left[\begin{array}{lll}

1 & 0 & 0 \\

0 & 1 & 0 \\

0 & 0 & 1

\end{array}\right]\) is an Identity or Unit matrix.

→ Zero Matrix or Null Matrix: A matrix, in which all the elements are equal to zero, is called the zero matrix.\(\left[\begin{array}{lll}

0 & 0 & 0 \\

0 & 0 & 0 \\

0 & 0 & 0

\end{array}\right]\) is a zero matrix.

→ Comparable Matrices: Two matrices are said to be comparable, if they are of the same order. For example, \(\left[\begin{array}{ccc}

2 & 3 & -5 \\

4 & -2 & 6

\end{array}\right]\) and \(\left[\begin{array}{ccc}

i & 2 & x \\

3 & x^{2} & -1

\end{array}\right]\) are comparable matrices since each matrix is of order 2 × 3.

→ Equal Matrices: Two matrices are equal, if (a) they are of the same order (b) their corresponding elements are equal.

if p = 2, q = 3, r = 5, s = 7, t = 9 and u = 8.

**OPERATIONS ON MATRICES:**

→ Addition of Matrices: The sum of two matrices A and B of the same order is obtained by adding the corresponding elements. Thus,

→ Multiplication of a Matrix by a Scalar: If a matrix A = [a_{ij}]_{m×n} is multiplied by a scalar k, then the product kA is obtained by multiplying each element of A, by k. For example,

→ Negative of a Matrix: The negative of a matrix A = -A = (-1)A. For example,

→ Difference of two Matrices: If A and B are the matrices of the same order, then A – B = A + (-1)B = Sum of matrices A and -B.

→ Properties of Matrices Addition: Let A, B, and C be the matrices of the same order m × n.

(a) The commutation Law: A + B = B + A

(b) The Association Law: (A + B) + C = A + (B + C)

(c) The Existence of Additive Identity: Let Omxn be null matrix of order m × n.

A + O_{m×n} = O_{m×n} + A = A.

(d) The Existence of Additive Inverse: Let A = [aij]m×n. We have an order matrix – A = [-a_{ij}]_{m×n} such that A + (-A) = A – A = O_{m×n}

-A is called the additive inverse of A or negative of A.

→ Properties of Scalar Multiplication of a Matrix

Let A and B be the matrices of the same order m × n. Then,

(a) k(A + B) = kA + kB

(b) (k + l)A = kA+ lA

→ Multiplication of Matrices

Two matrices A and B are conformable for multiplication if the Tiber of columns in A is equal to the number of rows in B.

If A = [a_{ij}]_{m×n} then B = [b_{ij}]_{n×p} and AB = [c_{ij}]_{m×p}

C_{ij} = (ij), the element of AB = sum of the products of the elements of the ith row of A with corresponding elements oi jth column of B. Here, (ith row of A) (jth column of B).

No. of columns in A = No. of rows in B = 2 ⇒ A and B are comformable for multiplication. Let AB = [C_{ij}]_{2×3}

c_{11} = (I row of A) × (I column of B)

= \(\left[\begin{array}{ll}

2 & 3

\end{array}\right]\left[\begin{array}{c}

-1 \\

2

\end{array}\right]\) = 2 × (-1) + 3 × 2 = -2 + 6 = 4

c_{12} = (I row of A) × (II column of B)

= \(\left[\begin{array}{ll}

2 & 3

\end{array}\right]\left[\begin{array}{c}

2 \\

-3

\end{array}\right]\) = 2 × 2 + 3 × (-3) = 4 – 9 = -5

c_{13} = (I row of A) × (III column of B)

= \(\left[\begin{array}{ll}

2 & 3

\end{array}\right]\left[\begin{array}{l}

4 \\

5

\end{array}\right]\) = 2 × 4 + 3 × 5 = 8 + 15 = 23

c_{21} = (II row of A) × (I column of B)

= \(\left[\begin{array}{ll}

1 & 4

\end{array}\right]\left[\begin{array}{c}

-1 \\

2

\end{array}\right]\) = 1 × (-1) + 4 × 2 = -1 + 8 = 7

c_{22} = (II row of A) × (III column of B)

= \(\left[\begin{array}{ll}

1 & 4

\end{array}\right]\left[\begin{array}{c}

2 \\

-3

\end{array}\right]\) = 1 × 2 + 4 × (-3) = 2 – 12 = -10

c_{23} = (II row of A) × (III column j B)

= \(\left[\begin{array}{ll}

1 & 4

\end{array}\right]\left[\begin{array}{l}

4 \\

5

\end{array}\right]\) =1 × 4 + 4 × 5 = 4 + 20 = 24

→ Properties of Multiplication of Matrices

(a) The Associative Law:

Let A = [a_{ij}]_{m×n}, B = [b_{ij}]_{n×p} and C = [c_{ij}]_{p×q}.

Then, (AB)C = A(BC)

(b) The Distributive Law:

- If A = [a
_{ij}]_{m×n}, B = [b_{ij}]_{n×p}and C = [c_{ij}]_{p×q}, then A(B + C) = AB + AC. - If A = [a
_{ij}]_{m×n}, B = [b_{ij}]_{n×p}and C = [c_{ij}]_{p×q}, then (A + B)C = AC + BC.

(c) The Existence of Multiplicative Identity:

Let A be a square matrix. There exists an identity matrix I of the same order such that IA = AI = A.

**TRANSPOSE OF A MATRIX:**

(a) Definition: Let A = [a_{ij}]_{m×n}. The matrix obtained by interchanging the rows and columns of A is called transpose of A. It is denoted by A’ or A^{T}.

For A = [a_{ij}]_{m×n} A’ = [aij]_{n×m}.

(b) Properties of Transpose of a Matrix Let A and B be the two matrices. Then,

- (A’)’ = A
- (kA)’ = kA’, where k is a scalar
- (A + B)’ = A’ + B’ (whenever A + B is defined)
- (AB)’ = B’A’ (whenever AB is defined)

**SYMMETRIC AND SKEW SYMMETRIC MATRICES:**

→ Symmetric Matrix: A square matrix A = [a_{ij}]_{n×n} is called symmetric, if A’ = A, i.e., for a_{ji} = a_{ij} e.g. \(\left[\begin{array}{ccc}

2 & 3 & 4 \\

3 & 1 & 5 \\

4 & 5 & -1

\end{array}\right]\) is a symmetric matrix.

→ Skew Symmetric Matrix: A square matrix A = [aij]n×n is skew symmetric, if A’ = -A for all i, j or a_{ji} = – a_{ij}\(\left[\begin{array}{ccc}

0 & 2 & -3 \\

-2 & 0 & 4 \\

3 & -4 & 0

\end{array}\right]\) a skew symmetric matrix.

→ Properties: Let A be a square matrix with real elements.

(a) A + A’ is symmetric.

(b) A – A’ is skew symrrtetric.

(c) A square matrix can be expressed as the sum of the symmetric and skew-symmetric matrix, i.e.,

A = \(\frac{1}{2}\) (A + A’) + \(\frac{1}{2}\) (A – A’)

**ELEMENTARY TRANSFORMATIONS OF A MATRIX:**

→ Interchange of ith row and jth row is denoted by R_{i} ↔ R_{j}. Similarly, interchange of the ith column with the jth column is denoted by C_{i} ↔ C_{j}.

→ Multiplication of each element of ith row by k is represented as R_{i} → kR_{j}. and when the ith column is multiplied by k, it is represented as C_{i} → kC_{j}.

→ Let the element of the ith row of A be added to the corresponding elements of the jth row multiplied by k. It is denoted by R_{i} → R_{i} + kR_{j}.

Similarly, in the case of columns when elements of the ith column are added to the corresponding elements of the jth column multiplied by k, then C_{i} → C_{j} + kC_{j}.

**INVERTIBLE MATRICES:**

(a) Definition: Let A be a square matrix of order n. If there exists another square matrix B of the same order such that AB = BA = I_{n}, then A is said to be invertible and B is called the inverse of A. It is denoted by A^{-1}. ⇒ B = A^{-1}.

(b) Inverse of a Matrix by Elementary Operations

Let B = A^{-1} be the inverse of A.

i.e., I_{n} = BA

Multiplying I_{n} by A^{-1}, we get

A^{-1}I_{n} = I_{n} A^{-1} = (BA)A^{-1} = B(AA^{-1})

= BI_{n} = B.

⇒ A^{-1} = B.

e.g. Let us find the inverse of \(\left[\begin{array}{ll}

1 & 2 \\

2 & 2

\end{array}\right]\) by elementary operations we have: \(\left[\begin{array}{ll}

1 & 2 \\

2 & 2

\end{array}\right]=\left[\begin{array}{ll}

1 & 0 \\

0 & 1

\end{array}\right]\) A

By elementary transformation, we change the matrix \(\left[\begin{array}{ll}

1 & 2 \\

2 & 2

\end{array}\right]\) so that we get identity matrix.

1. MATRIX

Def. A system of mn-numbers (real or complex) arranged in the form of an ordered set of m horizontal lines (called rows) and n vertical lines (called columns) is called an m x n matrix.

2. TYPES OF MATRICES

(i) Rectangular Matrix. Any m x n matrix (m≠n) is called a rectangular matrix.

(ii) Square Matrix. Any n x n matrix is called a square matrix of order n.

(iii) Row Matrix. Any 1 x n matrix is called a row matrix.

(iv) Column Matrix. Any m x 1 matrix is called a column matrix.

(v) Diagonal Matrix. A square matrix A = [a_{ij} ] is said to be a diagonal matrix if a_{ij} = 0 when i ≠ j.

(vi) Scalar Matrix. A diagonal matrix is said to be a scalar matrix if all its diagonal elements are equal.

(vii) Identity Matrix. A diagonal matrix is said to be an identity matrix if each of its diagonal elements is unity.

(viii) Zero Matrix. A matrix is said to be a zero matrix if each of its elements is zero.

(ix) Triangular Matrices.

(a) A square matrix A = [a_{ij} ] is said to be upper triangular matrix if a_{ij} =0 for i > j.

(b) A square matrix A = [a_{ij} ] is said to be lower triangular matrix if a_{ij} =0 for i < j.

3. EQUALITY OF MATRICES

Two matrices A = [a_{ij} ] and B = [b_{ij}] are said to be equal iff (i) they are of the same order (ii) their corresponding elements are equal.

4. OPERATIONS ON MATRICES

(i) Addition of Matrices.

Let A = [a_{ij}]_{m x n} and B = [b_{ij}]_{m x n} be two matrices. Then the sum A + B = C = [c_{ij}]_{m x n} , where c_{ij} + a_{ij} + b_{ij} for 1 ≤ i ≤ m, 1 ≤ j ≤ n

Key Point

Addition is defined only for matrices, which are of the same order.

(ii) Multiplication of a matrix by a scalar.

Let A be any m x n matrix and k be any scalar. Then m x n matrix obtained by multiplying each element by k is said to be scalar multiple of A by k and is denoted by kA or Ak.

(iii) Multiplication of Matrices.

Let A = [a_{ij}] be mxn matrix and B = [b_{jk}] be nxp matrix such that the number of columns of A equals the number of rows of B. Then matrix C = [c_{ik}] which is of the order m x p such that:

c_{ik} = \(\sum_{j=1}^{n} a_{i j} b_{j k}\) where i = 1,2, ….. m; p = 1, 2, 3,…………………k is called the product of the matrices A and B and is written as C = AB.

Key Point

Product AB is defined iff number of columns of A = number of rows of B.

5. TRANSPOSE OF A MATRIX

(i) Def. If A=[a_{ij}]_{m x n}, then the transpose of A, denoted by A’ (or A^{t} or A^{T}) is defined by n x m matrix obtained from A by writing the rows of A as columns and columns of A as rows in the same order.

(ii) Properties:

(a) (A’)’=A 1

(b) (A + B)’=A’+B’, A and B being of same type «

(c) (kA)’ = kA’, k being any scalar »

(d) (AB)’ = B’A’.

6. SYMMETRIC AND SKEW-SYMMETRIC MATRICES

(i) Symmetric Matrix.

Def. A square matrix A = [a_{ij}] is said to be symmetric if (i,j)th element is the same as its (j, i)th element.

Key Point is

A is symmetric if A’= A.

(ii) Skew-Symmetric Matrix.

A square matrix A = [a_{ij}] is said to be skew-symmetric if (i,j)th element is negative of its . (j, i)th element.

Key Point

A is skew-symmetric if A’ = – A.