Question

For any $2\times 2$ matrix A, if $A\left[ adj.A \right]=\left[ \begin{matrix} 10 & 0 \\\ 0 & 10 \\\ \end{matrix} \right],$ then |A| equals
(a) 0
(b) 10
(c) 20
(d) 100

Answer 1

To solve the given question, we will first find out what a matrix is and what adjoint of a matrix is. Then we will find the determinant of A and adjoint of A in terms of |A| which is given by $\left| adj\left( A \right) \right|={{\left| A \right|}^{n-1}}$. Then we will take the determinant of the LHS and RHS of the equation given in the question. In this, we will put the values of the determinants of A and adjoint of A in terms of |A|. From here, we will find the value of |A| which will be our required answer.

Complete step-by-step answer:
Before solving the given question, we will first find out what a matrix is and what adjoint of the matrix is. A matrix is a rectangular array or table of numbers, symbols or expressions, arranged in rows and columns. An adjoint of a matrix is the transpose of its cofactor matrix. The adjoint of a matrix X is denoted by adj(X).
Now, we will find the determinant of A. The determinant of A is represented by |A|. Now, we will find the determinant of the adjoint of A i.e. |Adj (A)|. For this, we have the following formula
$\left| adj\left( X \right) \right|={{\left| X \right|}^{n-1}}$
where n is the order of the matrix. In our case, n = 2. So, we have,
$\left| adj\left( A \right) \right|={{\left| A \right|}^{2-1}}$
$\Rightarrow \left| adj\left( A \right) \right|=\left| A \right|.....\left( i \right)$
Now, the equation given in the question is

10 & 0 \\\ 0 & 10 \\\ \end{matrix} \right]$$ Now, we will take the determinant on both the sides $$\left| A\left[ adj\left( A \right) \right] \right|=\left| \begin{matrix} 10 & 0 \\\ 0 & 10 \\\ \end{matrix} \right|$$ Now, we have the following property of a determinant. $$\left| XY \right|=\left| X \right|\left| Y \right|$$ Thus, we will get, $$\left| A \right|\left| adj\left( A \right) \right|=\left| \begin{matrix} 10 & 0 \\\ 0 & 10 \\\ \end{matrix} \right|.....\left( ii \right)$$ From (i), we will put the value of |adj A| in (ii). Thus, we will get, $$\left| A \right|\left| A \right|=\left| \begin{matrix} 10 & 0 \\\ 0 & 10 \\\ \end{matrix} \right|$$ $$\Rightarrow {{\left| A \right|}^{2}}=\left| \begin{matrix} 10 & 0 \\\ 0 & 10 \\\ \end{matrix} \right|$$ Now, the determinant $$\left| \begin{matrix} a & c \\\ b & d \\\ \end{matrix} \right|$$ has the value ad – bc. Thus, we have, $$\Rightarrow {{\left| A \right|}^{2}}=10\times 10-0\times 0$$ $$\Rightarrow {{\left| A \right|}^{2}}=100$$ $$\Rightarrow \left| A \right|=\sqrt{100}$$ $$\Rightarrow \left| A \right|=10$$ Hence, option (b) is correct. **Note:** The question given can also be solved in an alternate way as shown. We know that if A is a square matrix then we have the following relation. $$A\text{ }adj\left( A \right)=\left| A \right|{{I}_{n}}$$ where $${{I}_{n}}$$ is the unit matrix of the order n. In our case, n = 2. Thus, we have, $$A\text{ }adj\left( A \right)=\left| A \right|\left[ \begin{matrix} 1 & 0 \\\ 0 & 1 \\\ \end{matrix} \right]$$ $$\Rightarrow A\text{ }adj\left( A \right)=\left[ \begin{matrix} \left| A \right| & 0 \\\ 0 & \left| A \right| \\\ \end{matrix} \right]$$ On comparing this with the equation given in the question, we will get, $$\left| A \right|=10$$