ICSE Class 10 Maths Chapter 11 Matrices

Chapter 11

Matrices

Points To Remember

Matrix: A rectangular arrangement of numbers in the form of horizontal and vertical lines and enclosed by brackets [ ] or parenthesis ( ), is called a matrix. The horizontal lines in a matrix are called its rows and the vertical lines in it are called its columns. The numbers constituting a matrix are called its elements or entries. The plural of matrix is matrices.

Order of a Matrix: A matrix having m rows and n columns is called a matrix of order (m x n) read as (m by n).

Notation for the Elements of a Matrix: An element of a matrix, appearing in the ith row and j-th column is called the (i,j) element of the matrix and it is denoted by a subscript ij.

Diagonal Elements of a Matrix: In the given matrix, the (1,1)th, (2, 2)th (3, 3)th,... elements are called the diagonal elements. The line along which these elements lie is called the diagonal of the matrix.

Some Special Types Of Matrices

Row Matrix: A matrix having only one row is called a row matrix. Example: A = [2 5] and B = [7 0 - 3] are row matrices of order (1 x 2) and (1 x 3) respectively.

Column Matrix: A matrix having only one column is called a column matrix. Example: A = \[\begin{bmatrix} 4 \\ -1 \end{bmatrix}\] and B = \[\begin{bmatrix} -2 \\ 5 \\ 7 \end{bmatrix}\] are column matrices of order (2 x 1) and (3 x 1) respectively.

Rectangular Matrix: A Matrix in which the number of rows is not equal to the number of columns, is called a rectangular matrix. Example: A = \[\begin{bmatrix} 5 & 3 & 2 \\ 1 & -2 & 4 \end{bmatrix}\] is a rectangular matrix of order (2 x 3).

Square Matrix: A matrix in which the number of rows is equal to the number of columns, is called a square matrix. Example: A = \[\begin{bmatrix} 4 & 3 \\ 5 & 10 \end{bmatrix}\] is a square matrix of order 2. We may call it a 2-rowed square matrix.

Diagonal Matrix: A square matrix in which every non-diagonal element is 0 is called a diagonal matrix. Example: A = \[\begin{bmatrix} 5 & 0 \\ 0 & 9 \end{bmatrix}\] and B = \[\begin{bmatrix} 2 & 0 & 0 \\ 0 & 8 & 0 \\ 0 & 0 & 4 \end{bmatrix}\] are diagonal matrices of order (2 x 2) and (3 x 3) respectively.

Teacher's Note

Understanding matrix types helps organize data in real-world applications like spreadsheets and data tables you use in computer labs.

Unit or Identity Matrix: A square matrix in which every diagonal element is 1 and every non-diagonal element is 0, called a unit matrix. Example: \[\begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}\] and \[\begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix}\] are unit matrices of order 2 and 3 respectively.

Zero or Null Matrix: A matrix in which every element is 0, is called a zero or a null matrix. Example: [0 0], \[\begin{bmatrix} 0 \\ 0 \end{bmatrix}\] and \[\begin{bmatrix} 0 & 0 \\ 0 & 0 \end{bmatrix}\] are zero matrices of order (1 x 2), (2 x 1) and (2 x 2) respectively.

Comparable Matrices: Two matrices are said to be comparable, if they are of the same order. Example: The matrices A = \[\begin{bmatrix} 2 & -3 & 0 \\ 5 & 1 & -2 \end{bmatrix}\] and B = \[\begin{bmatrix} 5 & -7 & 2 \\ 8 & 0 & 6 \end{bmatrix}\] are comparable, since each one is of order (2 x 3).

Equal Matrices: Two matrices A and B are said to be equal, written as A = B, if they are of the same order and their corresponding elements are equal.

Operations On Matrices

Addition of Matrices: If A and B are two matrices of the same order, then their sum (A + B) is the matrix obtained by adding the corresponding elements of A and B.

Subtraction of Matrices: If A and B are two matrices of the same order, then their difference (A - B) is the matrix obtained by subtracting the elements of B from the corresponding elements of A.

Multiplication of a Matrix By a Number: If A is a given matrix and k is a real number, then the matrix kA is obtained by multiplying each element of A by k.

Some Properties Of Matrices

Let A, B and C be any matrices each order of (2 x 2) then, we have:

A + B = B + A (Commutative Law)

(A + B) + C = A + (B + C) (Associative Law)

Multiplication Of Matrices

For any two matrices A and B, the product AB exists only, when: Number of columns in A = Number of rows in B.

If A is an (m x n) matrix and B is an (n x p) matrix, then AB is a (m x p) matrix.

(i, k) th element of AB = (i-th row of A) x (k-th column of B).

Product Of Two Matrices Each Of Order (2 x 2)

If A and B are matrices each of order (2 x 2), then AB is a (2 x 2) matrix given by: AB = \[\begin{pmatrix} \text{1st row of A x 1st column of B} & \text{1st row of A x 2nd column of B} \\ \text{2nd row of A x 1st column of B} & \text{2nd row of A x 2nd column of B} \end{pmatrix}\]

Teacher's Note

Matrix multiplication is used in computer graphics to transform images and in engineering to solve systems of equations efficiently.

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