We have the matrix

A matrix, as we know, is a rectangular array of numbers, symbols, or expressions, arranged in rows and columns.

(i). We need to find the order of the matrix A.

And we know that,

The number of rows and columns that a matrix has is called its order or its dimension. By convention, rows are listed first; and columns second.

So,

Here, in matrix A:

There are 3 rows.

Elements in 1^st row = a, 1, x

Elements in 2^nd row = 2, √3, x² – y

Elements in 3^rd row = 0, 5, -2/5

⇒ M = 3

And,

There are 3 columns.

Elements in 1^st column = a, 2, 0

Elements in 2^nd column = 1, √3, 5

Elements in 3^rd column = x, x² – y, -2/5

⇒ N = 3

Since, the order of matrix = M × N

⇒ The order of matrix A = 3 × 3

Thus, the order of the matrix A is 3 × 3.

(ii). We need to find the number of elements in the matrix A.

And we know that,

Each number that makes up a matrix is called an element of the matrix.

So,

If a matrix has M rows and N columns, the number of elements is MN.

Here, in matrix A:

There are 3 rows.

⇒ M = 3

And,

There are 3 columns.

⇒ N = 3

Then, number of elements = MN

⇒ Number of elements = 3 × 3

⇒ Number of elements = 9

The elements are namely, a, 2, 0, 1, √3, 5, x, x² – y, -2/5.

Thus, the number of elements is 9.

(iii). We need to find the elements a₂₃, a₃₁ and a₁₂.

We know that,

a_ij is the representation of elements lying in the i^th row and j^th column.

For a₂₃:

Comparing a_ij with a₂₃, we have

i = 2

j = 3

Look up in matrix A for element in (i=) 2^nd row and (j=) 3^rd column.

Element that is common to both 2^nd row and 3^rd column = x² – y

⇒ a₂₃ = x² – y

For a₃₁:

Comparing a_ij with a₃₁, we have

i = 3

j = 1

Look up in matrix A for element in (i=) 3^rd row and (j=) 1^st column.

Element that is common to both 3^rd row and 1^st column = 0

⇒ a₃₁ = 0

For a₁₂:

Comparing a_ij with a₁₂, we have

i = 1

j = 2

Look up in matrix A for element in (i=) 1^st row and (j=) 2^nd column.

Element that is common to both 1^st row and 2^nd column = 1

⇒ a₁₂ = 1

Thus, a₂₃ = x² – y, a₃₁ = 0 and a₁₂ = 1.