If and A adj A=A AT, then

5 a + b is equal to :

 

  • Option 1)

    -1

  • Option 2)

    5

  • Option 3)

    4

  • Option 4)

    13

 

Answers (2)

As we learnt in 

Adjoint of a square matrix -

Transpose of the matrix of co-factors of elements of A is called the adjoint of A

- wherein

 

 

Transpose of a Matrix -

The matrix obtained from any given matrix A, by interchanging its rows and columns.

- wherein

 

 

A = \begin{bmatrix} 5a & -b\\ 3 & 2 \end{bmatrix}

\Rightarrow adj (A)= \begin{bmatrix} 2 & b\\ -3 & 5a \end{bmatrix}

\\ \Rightarrow A adj (A)= \begin{bmatrix} 5a & -b\\ 3 & 2 \end{bmatrix} \begin{bmatrix} 2 & b \\ -3 & 5a \end{bmatrix} = \begin{bmatrix} 10a + 3b & 0\\ 0 & 10a +3b \end{bmatrix}

\\ \Rightarrow A A ^{T}= \begin{bmatrix} 5a & -b\\ 3 & 2 \end{bmatrix} \begin{bmatrix} 5a & 3\\ -b & 2 \end{bmatrix}\\ = \begin{bmatrix} 25a^{2} + b^{2}& 15a-2b\\ 15a-2b & 13 \end{bmatrix}

so that 15a - 2b = 0

and 10a + 3b =13

\therefore a = \frac{2}{5} \ and \ b =3

\therefore 5a +b = 2+3 = 5

 


Option 1)

-1

This option is incorrect.

Option 2)

5

This option is correct.

Option 3)

4

This option is incorrect.

Option 4)

13

This option is incorrect.

N neha

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