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

Hint: 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
\[A\left[ adj\left( A \right) \right]=\left[ \begin{matrix}
   10 & 0 \\
   0 & 10 \\
\end{matrix} \right]\]
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,
\[\left| A \right|=10\]