ICSE Class 10 Maths Chapter 09 Matrices

Matrices

Matrix

A matrix is a rectangular arrangement of numbers, arranged in rows and columns.

For example: \(\begin{bmatrix}5\end{bmatrix}\), \(\begin{bmatrix}5 & 3\\1 & 2\end{bmatrix}\), \(\begin{bmatrix}5 & 3 & 2\end{bmatrix}\), etc.

Plural of matrix is matrices.

Each number or entity in a matrix is called its element.

In a matrix, the horizontal lines are called rows; whereas the vertical lines are called columns.

Order of a Matrix

The order of a matrix = Number of rows in it × Number of columns in it;

i.e. if a matrix has m number of rows and n number of columns, its order is written as m × n and is read as m by n.

Consider the matrix \(\begin{bmatrix}2 & 1 & 5\\3 & -2 & 7\end{bmatrix}\)

It has 2 rows and 3 columns; hence its order = 2 × 3 (read as 2 by 3)

While stating the order of a matrix, the number of rows is given first and then the number of columns.

Notation: Matrices, in general, are denoted by capital letters. For example, if A is a matrix with m rows and n columns, then it is written as A_{m × n}.

Similarly, B_{5 × 3} means, a matrix B with 5 rows and 3 columns.

Elements of a Matrix

Each number or entity in a matrix is called its element.

The total number of elements in a matrix is equal to the product of its number of rows and number of columns, i.e. if a matrix has 4 rows and 6 columns, the number of elements in it = 4 × 6 = 24.

Consider matrix A = \(\begin{bmatrix}2 & 4 & -3\\0 & 1 & 2\end{bmatrix}\)

Since, matrix A has 2 rows and 3 columns, so the number of elements in it = 2 × 3 = 6.

It must be noted here that if a matrix has 6 elements, then it may have:

(i) 1 row and 6 columns; as 1 × 6 = 6, or

(ii) 2 rows and 3 columns; as 2 × 3 = 6, or

(iii) 3 rows and 2 columns; as 3 × 2 = 6, or

(iv) 6 rows and 1 column; as 6 × 1 = 6.

Similarly, if a matrix has 8 elements, it may have:

(i) 1 row and 8 columns so that its order = 1 × 8 and number of elements in it = 1 × 8 = 8, or

(ii) 2 rows and 4 columns so that its order = 2 × 4 and number of elements in it = 2 × 4 = 8, or

(iii) 4 rows and 2 columns so that its order = 4 × 2 and number of elements in it = 4 × 2 = 8, or

(iv) 8 rows and 1 column so that its order = 8 × 1 and number of elements in it = 8 × 1.

Types of Matrices

Row Matrix: A matrix which has only one row is called a row matrix.

For example: \(\begin{bmatrix}a & b\end{bmatrix}\)

Since, this matrix has 1 row and 2 columns, its order = 1 × 2 (1 by 2).

Similarly, \(\begin{bmatrix}a & b & c\end{bmatrix}\) is a row matrix of order 1 × 3.

A row matrix is also called a row vector.

Column Matrix: A matrix which has only one column is called a column matrix.

For example: \(\begin{bmatrix}a\\b\end{bmatrix}\)

Since, this matrix has 2 rows and 1 column, its order = 2 × 1 (2 by 1).

Similarly, \(\begin{bmatrix}a\\b\\c\end{bmatrix}\) is a column matrix of order 3 × 1.

A column matrix is also called a column vector.

Square Matrix: A matrix which has an equal number of rows and columns is called a square matrix.

For example: \(\begin{bmatrix}a & b\\c & d\end{bmatrix}\)

Since, this matrix has 2 rows and 2 columns, its order = 2 × 2 (2 by 2).

Similarly, \(\begin{bmatrix}5 & 7 & 4\\2 & -1 & 0\\0 & 3 & 4\end{bmatrix}\) is a square matrix of order 3 × 3.

Rectangular Matrix: A matrix in which the number of rows are not equal to the number of columns is called a rectangular matrix.

For example: \(\begin{bmatrix}2 & 4 & 7\\1 & 0 & 5\end{bmatrix}\) and \(\begin{bmatrix}3 & 1\\6 & 2\\1 & 7\end{bmatrix}\)

Order is 2 × 3 and Order is 3 × 2

Zero or Null Matrix: If each element of a matrix is zero, it is called a zero matrix or a null matrix.

For example: \(\begin{bmatrix}0 & 0\end{bmatrix}\), \(\begin{bmatrix}0\\0\end{bmatrix}\), \(\begin{bmatrix}0 & 0\\0 & 0\end{bmatrix}\), etc.

Diagonal Matrix: A square matrix which has all its elements zero each except those on the leading (or, principal) diagonal is called a diagonal matrix.

For example: \(\begin{bmatrix}2 & 0\\0 & 3\end{bmatrix}\), \(\begin{bmatrix}5 & 0 & 0\\0 & -2 & 0\\0 & 0 & 3\end{bmatrix}\), etc.

In a square matrix, the leading (principal) diagonal means the diagonal from top left to bottom right.

Unit or Identity Matrix: A diagonal matrix in which each element of its leading diagonal is unity (i.e. 1) is called a unit or identity matrix. It is denoted by I. In other words, it is a square matrix in which each element of its leading diagonal is equal to 1 and all other remaining elements of the matrix are zero each.

For example: \(\begin{bmatrix}1 & 0\\0 & 1\end{bmatrix}\), \(\begin{bmatrix}1 & 0 & 0\\0 & 1 & 0\\0 & 0 & 1\end{bmatrix}\), etc.

Transpose of a Matrix

Transpose of a matrix is the matrix obtained on interchanging its rows and columns. If A is a matrix, then its transpose is denoted by A^t.

For example: If A = \(\begin{bmatrix}2 & 3 & 1\\0 & 4 & 7\end{bmatrix}\), then its transpose A^t = \(\begin{bmatrix}2 & 0\\3 & 4\\1 & 7\end{bmatrix}\)

Equality of Matrices

Two matrices are said to be equal if:

(i) both the matrices have the same order,

(ii) the corresponding elements of both the matrices are equal.

i.e. if A = \(\begin{bmatrix}2 & 3\\1 & 5\end{bmatrix}\) and B = \(\begin{bmatrix}2 & 3\\1 & 5\end{bmatrix}\); then A = B.

Teacher's Note

Matrices are used in computer graphics to rotate, scale, and translate images on your screen. Every time you play a video game or scroll through social media, matrices are working behind the scenes.

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