Question

If A is a unit matrix of order n, then which of the following is true for \[A\left( {adj\,A} \right)\]?
A. Zero matrix
B. Row matrix
C. Unit matrix
D. None of these

Answer 1

Hint: To solve this question we will use the formula of a matrix inverse. We will multiply A on both sides of the formula and then solve the equation to get the required solution.

Formula used:
The matrix inverse formula is \[{A^{ - 1}} = \dfrac{{Adj\,A}}{{\left| A \right|}}\].
The product of a matrix with its inverse is an identity matrix.
\[A\dot A^{-1} = I\]

Complete step by step solution:
We know that the formula of the inverse matrix is
\[{A^{ - 1}} = \dfrac{{Adj\,A}}{{\left| A \right|}}\]
Multiply A on both sides of the above equation
\[ \Rightarrow A{A^{ - 1}} = A\dfrac{{Adj\,A}}{{\left| A \right|}}\]
We know that the product of a matrix with its inverse is an identity matrix. Thus \[A{A^{ - 1}} = I\].
\[ \Rightarrow I = A\dfrac{{Adj\,A}}{{\left| A \right|}}\]
Multiply both sides by \[\left| A \right|\].
\[ \Rightarrow I \cdot \left| A \right| = A\dfrac{{Adj\,A}}{{\left| A \right|}} \cdot \left| A \right|\]
Since A is unit matrix, \[\left| A \right| = 1\]
Substitute \[\left| A \right| = 1\] in the above equation:
\[ \Rightarrow I \cdot 1 = A\left( {Adj\,A} \right)\]
\[ \Rightarrow I = A\left( {Adj\,A} \right)\]
Thus \[A\left( {adj\,A} \right)\] is an identity matrix. The determinant value of an identity matrix is 1.
So, an identity matrix is also called a unit matrix.
Thus, \[A\left( {adj\,A} \right)\] is a unit matrix.
Hence option C is the correct option.

Note:To solve the given question, we must know about the formula of the inverse matrix. Using this relation we can find out the result. From the formula, we will find the value of \[A\left( {adj\,A} \right)\] then we will identity which type of matrix it represents.