Write the adjoint of the matrix. 2 - 1 4 3 .

Let A = 2 - 1 4 3 = a 11 a 12 a 21 a 22 Co-factors of elements of coefficient matrix A are as follows: C 11 = - 1 1 + 1 M 11 = 1 × 3 = 3 C 12 = - 1 1 + 2 M 12 = - 1 × 4 = - 4 C 21 = - 1 2 + 1 M 21 = - 1 × - 1 = 1 C 22 = - 1 2 + 2 M 22 = 1 × 2 = 2 Therefore, the Co-factor matrix C of the matrix A is C = 3 - 4 1 2 Adj A = Transpose of C 11 C 12 C 21 C 22 Adj A = 3 - 4 1 2 ' = 3 1 - 4 2 . This is the required adjoint of given matrix.

Write the adjoint of the matrix. 2 - 1 4 3 .

Let A = 2 - 1 4 3 = a 11 a 12 a 21 a 22 Co-factors of elements of coefficient matrix A are as follows: C 11 = - 1 1 + 1 M 11 = 1 × 3 = 3 C 12 = - 1 1 + 2 M 12 = - 1 × 4 = - 4 C 21 = - 1 2 + 1 M 21 = - 1 × - 1 = 1 C 22 = - 1 2 + 2 M 22 = 1 × 2 = 2 Therefore, the Co-factor matrix C of the matrix A is C = 3 - 4 1 2 Adj A = Transpose of C 11 C 12 C 21 C 22 Adj A = 3 - 4 1 2 ' = 3 1 - 4 2 . This is the required adjoint of given matrix.

Write the adjoint of the matrix. 3 0 - 2 - 5 .

Let A = 3 0 - 2 - 5 = a 11 a 12 a 21 a 22 Co-factors of elements of coefficient matrix A are as follows: C 11 = - 1 1 + 1 M 11 = 1 × ( - 5 ) = - 5 C 12 = - 1 1 + 2 M 12 = - 1 × ( - 2 ) = 2 C 21 = - 1 2 + 1 M 21 = - 1 × 0 = 0 C 22 = - 1 2 + 2 M 22 = 1 × ( 3 ) = 3 Therefore, the Co-factor matrix C of the matrix A is C = - 5 2 0 3 Adj A = Transpose of C 11 C 12 C 21 C 22 Adj A = - 5 2 0 3 ' = - 5 0 2 3 . This is the required adjoint of given matrix.

For what value of k matrix A = 2 k 3 5 has no inverse?

We have to find the value of k . Given matrix A = 2 k 3 5 A square matrix has no inverse if and only if it is a singular matrix. A matrix is singular matrix if its determinant is zero i . e . , A = 0 ⇒ 2 k 3 5 = 0 ⇒ 2 × 5 - 3 × k = 0 ⇒ 10 - 3 k = 0 ⇒ 3 k = 10 ⇒ k = 10 3 . When k = 10 3 then A has no inverse.

Write A - 1 for A = 2 5 1 3 .

Given matrix A = 2 5 1 3 Then A = 2 5 1 3 = ( 2 × 3 ) - ( 5 × 1 ) = 6 - 5 = 1 ≠ 0 So, A is a non-singular matrix and therefore it is invertible. Let C ij be cofactor of a ij in A . Then, the cofactors of elements of A are given by C 11 = 3 , C 12 = - 1 , C 21 = - 5 , C 22 = 2 ∴ Adj A = C T = C 11 C 12 C 21 C 22 T = 3 - 1 - 5 2 T = 3 - 5 - 1 2 Hence, A - 1 = 1 A adjA = 1 1 3 - 5 - 1 2 = 3 - 5 - 1 2 . This is the required inverse of given matrix.

If A = 3 10 2 7 , then write A - 1 .

Given matrix A = 3 10 2 7 Then A = 3 10 2 7 = ( 3 × 7 ) - ( 10 × 2 ) = 21 - 20 = 1 ≠ 0 So, A is a non-singular matrix and therefore it is invertible. Let C ij be cofactor of a ij in A . Then, the cofactors of elements of A are given by C 11 = 7 , C 12 = - 2 , C 21 = - 10 , C 22 = 3 ∴ adjA = 7 - 2 - 10 3 T = 7 - 10 - 2 3 Hence, A - 1 = 1 A adjA = 1 1 7 - 10 - 2 3 = 7 - 10 - 2 3 This is the required inverse of given matrix.

Write the inverse of the matrix 3 - 1 1 2 .

Given matrix A = 3 - 1 1 2 Then A = 3 - 1 1 2 = ( 3 × 2 ) - ( ( - 1 ) × 1 ) = 6 + 1 = 7 ≠ 0 So, A is a non-singular matrix and therefore it is invertible. Let C ij be cofactor of a ij in A . Then, the cofactors of elements of A are given by C 11 = 2 , C 12 = - 1 , C 21 = 1 , C 22 = 3 ∴ adjA = 2 - 1 1 3 T = 2 1 - 1 3 Hence, A - 1 = 1 A adjA = 1 7 2 1 - 1 3 This is the required inverse of given matrix.