The number of matrices
\(A=\begin{pmatrix} a & b \\ c & d \\ \end{pmatrix}\), where a,b,c,d ∈−1,0,1,2,3,…..,10
such that A = A-1, is ______.
The number of matrices
\(A=\begin{pmatrix} a & b \\ c & d \\ \end{pmatrix}\), where a,b,c,d ∈−1,0,1,2,3,…..,10
such that A = A-1, is ______.
Correct Answer: 50
Solution and Explanation
The correct answer is 50
\(∵ A=\begin{bmatrix} a & b \\ c & d \\ \end{bmatrix}\) then \(A^2=\begin{bmatrix} a^2+bc & b(a+d) \\ c(a+d) & bc+d^2 \\ \end{bmatrix}\)
For A–1 must exist ad – bc≠ 0 …(i)
and A = A–1⇒A2 = I
∴ a2 + bc = d2 + bc = 1 …(ii)
and b(a + d) = c(a + d) = 0 …(iii)
Case I : When a = d = 0, then possible values of
(b, c) are (1, 1), (–1, 1) and (1, –1) and (–1, 1).
Total four matrices are possible.
Case II : When a = –d then (a, d) be (1, –1) or
(–1, 1).
Then total possible values of (b, c) are
(12 + 11) × 2 = 46.
∴ Total possible matrices = 46 + 4 = 50.
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Concepts Used:
Matrices
Matrix:
A matrix is a rectangular array of numbers, variables, symbols, or expressions that are defined for the operations like subtraction, addition, and multiplications. The size of a matrix is determined by the number of rows and columns in the matrix.
The basic operations that can be performed on matrices are:
- Addition of Matrices - The addition of matrices addition can only be possible if the number of rows and columns of both the matrices are the same.
- Subtraction of Matrices - Matrices subtraction is also possible only if the number of rows and columns of both the matrices are the same.
- Scalar Multiplication - The product of a matrix A with any number 'c' is obtained by multiplying every entry of the matrix A by c, is called scalar multiplication.
- Multiplication of Matrices - Matrices multiplication is defined only if the number of columns in the first matrix and rows in the second matrix are equal.
- Transpose of Matrices - Interchanging of rows and columns is known as the transpose of matrices.