CBSE Class 12 Mathematics Matrices Notes Set 02

Important Notes and Formulas

Definition

A set of mn numbers arranged in the form of an ordered set of m rows and n columns is called an m × n matrix (to be read as m by n matrix).
This m × n matrix A is written as
\( A = \begin{bmatrix} a_{11} & a_{12} & \cdots & a_{1n} \\ a_{21} & a_{22} & \cdots & a_{2n} \\ a_{31} & a_{32} & \cdots & a_{3n} \\ \vdots & \vdots & \ddots & \vdots \\ a_{m1} & a_{m2} & \cdots & a_{mn} \end{bmatrix} \)
or \( A = [a_{ij}] \);   i = 1, 2, …, m;  j = 1, 2, …, n
or \( A = [a_{ij}]_{m \times n} \)
where \( a_{ij} \) represents the element at the intersection of the i^th row and j^th column.
In case the order of a matrix is established or known, then we shall simply write
\( A = [a_{ij}] \) of type m × n.

Types of Matrices

(1) Row Matrix
A matrix having only one row is called a row matrix or a row vector. For example,
A = [1  2  −1  −2] is a row matrix of order 1 × 4.

(2) Column Matrix
A matrix having only one column is called a column matrix or a column vector.
For example, \( A = \begin{bmatrix} 1 \\ 2 \\ -1 \end{bmatrix} \) and \( B = \begin{bmatrix} 3 \\ 2 \\ 5 \\ 4 \end{bmatrix} \) are column matrices of order 3 × 1 and 4 × 1 respectively.

(3) Square Matrix
A matrix in which the number of rows is equal to the number of columns, say (n, n), is called a square matrix of order n.
A square matrix of order n is called an n-rowed square matrix.
The elements \( a_{ij} \) of a square matrix \( A = [a_{ij}]_{n \times n} \) for which i = j, i.e. the elements \( a_{11}, a_{22}, \ldots, a_{nn} \) are called the diagonal elements and the line along which they lie is called the principal diagonal or leading diagonal of the matrix.
For example, the matrix \( \begin{bmatrix} 2 & 1 & -1 \\ 3 & -2 & 5 \\ 1 & 5 & -3 \end{bmatrix} \) is a square matrix of order 3 in which the diagonal elements are 2, −2, −3.

(4) Diagonal Matrix
A square matrix \( A = [a_{ij}]_{n \times n} \) is called a diagonal matrix if all the elements, except those in the leading diagonal, are zero, i.e. \( a_{ij} = 0 \) for all i ≠ j.
A diagonal matrix of order n × n having \( d_1, d_2, \ldots, d_n \) as diagonal elements is denoted by diag \( [d_1, d_2, \ldots, d_n] \).
Example: The matrix \( A = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 3 \end{bmatrix} \) is a diagonal matrix, to be denoted by A = diag [1, 2, 3].

(5) Scalar Matrix
A square matrix \( A = [a_{ij}]_{n \times n} \) is called a scalar matrix if
(i) \( a_{ij} = 0 \) for all i ≠ j and
(ii) \( a_{ii} = C \) for all i, where C ≠ 0.
In other words, a diagonal matrix in which all the diagonal elements are equal is called a scalar matrix.
Example: The matrices \( A = \begin{bmatrix} 2 & 0 \\ 0 & 2 \end{bmatrix} \), \( B = \begin{bmatrix} 1-2i & 0 & 0 \\ 0 & 1-2i & 0 \\ 0 & 0 & 1-2i \end{bmatrix} \) are scalar matrices of order 2 and 3 respectively.

(6) Identity or Unit Matrix
A square matrix \( A = [a_{ij}]_{n \times n} \) is called an identity or unit matrix if
(i) \( a_{ij} = 0 \) for all i ≠ j, and
(ii) \( a_{ii} = 1 \) for all i.
In other words, a square matrix each of whose diagonal elements is unity and each of whose non-diagonal elements is equal to zero is called an identity or unit matrix.
The identity matrix of order n is denoted by \( I_n \).
Example: The matrices \( I_2 = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} \), \( I_3 = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix} \) are identity matrices of order 2 and 3 respectively.

(7) Null Matrix
A matrix whose all elements are zero is called a null matrix or a zero matrix.
Example: \( \begin{bmatrix} 0 & 0 \\ 0 & 0 \end{bmatrix} \), \( \begin{bmatrix} 0 & 0 & 0 \\ 0 & 0 & 0 \end{bmatrix} \) are null matrices of order 2 × 2 and 2 × 3 respectively.

Equality of Matrices

Two matrices \( A = [a_{ij}]_{m \times n} \) and \( B = [b_{ij}]_{r \times s} \) are equal if
(i) m = r, i.e. the number of rows in A equals the number of rows in B.
(ii) n = s, i.e. the number of columns in A equals the number of columns in B.
(iii) \( a_{ij} = b_{ij} \) for i = 1, 2, …, m and j = 1, 2, …, n.
If two matrices A and B are equal, we write A = B, otherwise we write A ≠ B.
e.g. the matrices \( \begin{bmatrix} 0 & 0 \\ 0 & 0 \end{bmatrix} \) and \( \begin{bmatrix} 0 & 0 & 0 \\ 0 & 0 & 0 \end{bmatrix} \) are not equal, because their orders are not the same.

Sum of Matrices

Let \( A = [a_{ij}] \), \( B = [b_{ij}] \) be two matrices of the same type m × n. Then their sum (or difference) A + B (or A − B) is defined as another matrix of the same type, say \( C = [c_{ij}] \), such that any element of C is the sum (or difference) of the corresponding elements of A and B.
∴ \( C = A \pm B = [a_{ij} \pm b_{ij}] \)
e.g. \( A = \begin{bmatrix} 1 & 2 & 4 \\ 0 & 5 & 3 \end{bmatrix} \) and \( B = \begin{bmatrix} 7 & 3 & 2 \\ 5 & 1 & 9 \end{bmatrix} \)
Here both A and B are 2 × 3 matrices.
∴ \( A + B = \begin{bmatrix} 1+7 & 2+3 & 4+2 \\ 0+5 & 5+1 & 3+9 \end{bmatrix} = \begin{bmatrix} 8 & 5 & 6 \\ 5 & 6 & 12 \end{bmatrix} \)
and \( A - B = \begin{bmatrix} 1-7 & 2-3 & 4-2 \\ 0-5 & 5-1 & 3-9 \end{bmatrix} = \begin{bmatrix} -6 & -1 & 2 \\ -5 & 4 & -6 \end{bmatrix} \)

Properties of Matrix Addition
(i) Matrix addition is commutative: A + B = B + A
(ii) Matrix addition is associative: A + (B + C) = (A + B) + C

Negative of a Matrix: If A is a given matrix, then −A is called the negative of matrix A and all its elements are the corresponding elements of A multiplied by −1.
Thus if \( A = \begin{bmatrix} 2 & 3 & -1 \\ 6 & -4 & 2 \end{bmatrix} \), then \( -A = \begin{bmatrix} -2 & -3 & 1 \\ -6 & 4 & -2 \end{bmatrix} \)

Scalar Multiple of a Matrix

If A is a given matrix and k is any scalar number, real or complex [we call it scalar k to distinguish it from the matrix [k], which is a 1 × 1 matrix], then kA = Ak. It means all the elements of kA are k times the corresponding elements of A.
If \( A = \begin{bmatrix} 2 & 3 & 1 \\ 5 & 2 & 4 \end{bmatrix} \), then
\( 3A = \begin{bmatrix} 3 \cdot 2 & 3 \cdot 3 & 3 \cdot 1 \\ 3 \cdot 5 & 3 \cdot 2 & 3 \cdot 4 \end{bmatrix} \) or \( 3A = \begin{bmatrix} 6 & 9 & 3 \\ 15 & 6 & 12 \end{bmatrix} \)
Similarly \( -4A = \begin{bmatrix} -4 \cdot 2 & -4 \cdot 3 & -4 \cdot 1 \\ -4 \cdot 5 & -4 \cdot 2 & -4 \cdot 4 \end{bmatrix} = \begin{bmatrix} -8 & -12 & -4 \\ -20 & -8 & -16 \end{bmatrix} \)

Matrix Multiplication

If \( A = [a_{ij}]_{m \times p} \) and \( B = [b_{jk}]_{p \times n} \), then
\( [A]_{m \times p} \times [B]_{p \times n} = [AB]_{m \times n} \)
\( C = AB = [C_{ik}]_{m \times n} \), where \( C_{ik} = \sum_{j=1}^{p} a_{ij} b_{jk} \)
For multiplication, the number of columns of the first matrix should be equal to the number of rows of the second matrix.
i.e. \( C_{ik} = a_{i1} b_{1k} + a_{i2} b_{2k} + \ldots + a_{ip} b_{pk} \)
In other words, \( C_{ik} \) = sum of the products of the elements of the i^th row of A (having p elements) with the corresponding elements of the k^th column of B (having p elements). This is known as row by column multiplication of matrices.

Properties of Matrix Multiplication
(a) Multiplication of matrices is distributive with respect to addition of matrices, i.e. A(B + C) = AB + AC.
(b) Matrix multiplication is associative if conformability is assured, i.e. A(BC) = (AB)C.
(c) The multiplication of matrices is not always commutative, i.e. AB is not always equal to BA.
(d) Multiplication of a matrix A by a null matrix conformable with A for multiplication is a null matrix, i.e. AO = O. In particular, if A is a square matrix and O is the square null matrix of the same order, then OA = AO = O.
(e) If AB = O, then it does not necessarily mean that A = O or B = O or both are O, as shown below.
\( \begin{bmatrix} 0 & 0 \\ 0 & 1 \end{bmatrix} \begin{bmatrix} 1 & 0 \\ 0 & 0 \end{bmatrix} = \begin{bmatrix} 0 & 0 \\ 0 & 0 \end{bmatrix} \)
None of the matrices on the left is a null matrix, whereas their product is a null matrix.
(f) Multiplication of a matrix by a unit matrix I: Let A be an m × n matrix and I be a square unit matrix of order n, so that A and I are conformable for multiplication; then \( AI_n = A \).
Similarly, for IA to exist, I should be a square unit matrix of order m and in that case \( I_m A = A \).

Zero Divisors: If A and B are non-zero matrices such that AB = O, then A and B are called divisors of zero.
\( A = \begin{bmatrix} 0 & c & -b \\ -c & 0 & a \\ b & -a & 0 \end{bmatrix} \);  \( B = \begin{bmatrix} a^2 & ab & ac \\ ab & b^2 & bc \\ ac & bc & c^2 \end{bmatrix} \)
\( AB = \begin{bmatrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{bmatrix} = O \),   A ≠ O, B ≠ O.

AB = AC does not imply B = C, i.e. the cancellation law does not hold for matrix multiplication.
\( A = \begin{bmatrix} 1 & -3 & 2 \\ 2 & 1 & -3 \\ 4 & -3 & -1 \end{bmatrix} \);  \( B = \begin{bmatrix} 1 & 4 & 1 & 0 \\ 2 & 1 & 1 & 1 \\ 1 & -2 & 1 & 2 \end{bmatrix} \);  \( C = \begin{bmatrix} 2 & 1 & -1 & -2 \\ 3 & -2 & -1 & -1 \\ 2 & -5 & -1 & 0 \end{bmatrix} \)
\( AB = AC = \begin{bmatrix} -3 & -3 & 0 & 1 \\ 1 & 15 & 0 & -5 \\ -3 & 15 & 0 & -5 \end{bmatrix} \)
But B ≠ C, A ≠ O.

The Transpose of a Matrix

If A is a given matrix of the type m × n, then the matrix obtained by changing the rows of A into columns and the columns of A into rows is called the transpose of matrix A and is denoted by A′. There are m rows in A, therefore there will be m columns in A′, and similarly as there are n columns in A there will be n rows in A′. Hence the matrix A′ is of type n × m.
e.g. \( A = \begin{bmatrix} 3 & 4 \\ 2 & 1 \\ 5 & 9 \end{bmatrix}_{3 \times 2} \), then \( A^T = A' = \begin{bmatrix} 3 & 2 & 5 \\ 4 & 1 & 9 \end{bmatrix}_{2 \times 3} \)

Properties of Transpose
(1) (A′)′ = A
(2) (KA)′ = KA′, K being a scalar
(3) (A + B)′ = A′ + B′
(4) (AB)′ = B′A′
(5) (ABC)′ = C′B′A′

Symmetric Matrices

A square matrix \( A = [a_{ij}] \) will be called symmetric if for all values of i and j, \( a_{ij} = a_{ji} \), i.e. when \( A^T = A \).
i.e. every (i, j)^th element = (j, i)^th element.
e.g. \( A = \begin{bmatrix} a & h & g \\ h & b & f \\ g & f & c \end{bmatrix}_{3 \times 3} \)

Skew Symmetric Matrix

A square matrix \( A = [a_{ij}] \) will be called skew symmetric if its (i, j)^th element is the negative of its (j, i)^th element for all values of i and j, i.e. \( a_{ij} = -a_{ji} \) for all values of i and j, i.e. when \( A^T = -A \).
Since diagonal elements will be of the type \( a_{11}, a_{22}, a_{33}, \ldots, a_{ii} \), by the given condition \( a_{ii} = -a_{ii} \) for all values of i [i = j when elements are diagonal]
or \( 2a_{ii} = 0 \)   ∴ \( a_{ii} = 0 \)
Hence the diagonal elements of a skew symmetric matrix are zero.
e.g. \( \begin{bmatrix} 0 & h & g \\ -h & 0 & f \\ -g & -f & 0 \end{bmatrix} \) is a skew symmetric matrix.

Solved Examples

Example 1. If \( \begin{bmatrix} x-y & 2x+z \\ 2x-y & 3z+w \end{bmatrix} = \begin{bmatrix} -1 & 5 \\ 0 & 13 \end{bmatrix} \), find x, y, z, w.
Answer:
Since the corresponding elements of two equal matrices are equal, therefore
\( \begin{bmatrix} x-y & 2x+z \\ 2x-y & 3z+w \end{bmatrix} = \begin{bmatrix} -1 & 5 \\ 0 & 13 \end{bmatrix} \)
⇒ x − y = −1, 2x + z = 5, 2x − y = 0, 3z + w = 13.
Solving the equations x − y = −1 and 2x − y = 0 as simultaneous linear equations, we get x = 1, y = 2. Now putting x = 1 in 2x + z = 5, we get z = 3. Substituting z = 3 in 3z + w = 13, we obtain w = 4.

Example 2. Find the values of x, y, z and a which satisfy the matrix equation \( \begin{bmatrix} x+3 & 2y+x \\ z-1 & 4a-6 \end{bmatrix} = \begin{bmatrix} 0 & -7 \\ 3 & 2a \end{bmatrix} \).
Answer:
Since the corresponding elements of two equal matrices are equal, therefore
x + 3 = 0, 2y + x = −7, z − 1 = 3 and 4a − 6 = 2a.
Solving these equations, we get
a = 3, x = −3, y = −2, z = 4.

Example 3. If \( A = \begin{bmatrix} 1 & -2 & 3 \\ -4 & 2 & 5 \end{bmatrix}_{2 \times 3} \) and \( B = \begin{bmatrix} 2 & 3 \\ 4 & 5 \\ 2 & 1 \end{bmatrix}_{3 \times 2} \), compute AB and show that AB ≠ BA.
Answer:
A is of type 2 × 3 and B is of type 3 × 2, and hence both AB and BA are defined because the number of columns in the pre-factor is equal to the number of rows in the post-factor.
\( AB = \begin{bmatrix} 1 \cdot 2 + (-2) \cdot 4 + 3 \cdot 2 & 1 \cdot 3 + (-2) \cdot 5 + 3 \cdot 1 \\ (-4) \cdot 2 + 2 \cdot 4 + 5 \cdot 2 & (-4) \cdot 3 + 2 \cdot 5 + 5 \cdot 1 \end{bmatrix} = \begin{bmatrix} 0 & -4 \\ 10 & 3 \end{bmatrix}_{2 \times 2} \)
\( BA = \begin{bmatrix} 2 & 3 \\ 4 & 5 \\ 2 & 1 \end{bmatrix}_{3 \times 2} \begin{bmatrix} 1 & -2 & 3 \\ -4 & 2 & 5 \end{bmatrix}_{2 \times 3} = \begin{bmatrix} -10 & 2 & 21 \\ -16 & 2 & 37 \\ -2 & -2 & 11 \end{bmatrix}_{3 \times 3} \)
Hence AB ≠ BA.