Question

If A is a square matrix of order 3 and \[{{\rm{A}}^{\rm{T}}}\] denotes the transpose of matrix A, \[{{\rm{A}}^{\rm{T}}}{\rm{A = I}}\] and \[{\rm{det A = 1}}\], then \[{\rm{det (A}} - {\rm{I)}}\] must be equal to

Answer 1

Hint:
Here, in this question, we have to use the basic concept of the matrix to find the value of\[{\rm{det (A}} - {\rm{I)}}\]. Firstly we will write all the cases of the possible matrix of A as the determinant of matrix A is 1 and then finding the value of\[{\rm{det (A}} - {\rm{I)}}\] for each case.

Complete step by step solution:
So it is given that the determinant of matrix A is 1 i.e. \[{\rm{det A = 1}}\]and we know that the identity matrix is the matrix whose determinant is equal to 1.
So, matrix A will be an identity matrix in which the rows and columns can be interchanged. Therefore there are 3 valid cases of matrix A.
First case when matrix A is \[\left( {\begin{array}{*{20}{c}}
1&0&0\\
0&1&0\\
0&0&1
\end{array}} \right)\]
Second case when matrix A is \[\left( {\begin{array}{*{20}{c}}
0&1&0\\
0&0&1\\
1&0&0
\end{array}} \right)\]
Third case when matrix A is \[\left( {\begin{array}{*{20}{c}}
0&0&1\\
1&0&0\\
0&1&0
\end{array}} \right)\]
Now we will u the value of matrix A in\[{\rm{det (A}} - {\rm{I)}}\] to find its value.
For the first case when matrix A is \[\left( {\begin{array}{*{20}{c}}
1&0&0\\
0&1&0\\
0&0&1
\end{array}} \right)\]
\[{\rm{det (A}} - {\rm{I)}} = \left| {\left( {\begin{array}{*{20}{c}}
1&0&0\\
0&1&0\\
0&0&1
\end{array}} \right) - \left( {\begin{array}{*{20}{c}}
1&0&0\\
0&1&0\\
0&0&1
\end{array}} \right)} \right| = \left| {\left( {\begin{array}{*{20}{c}}
0&0&0\\
0&0&0\\
0&0&0
\end{array}} \right)} \right| = 0\]
Similarly, we will find the value of\[{\rm{det (A}} - {\rm{I)}}\]for the other two cases of matrix A and for all the cases of matrix A value of\[{\rm{det (A}} - {\rm{I)}}\]is 0.

Hence, 0 is the value of the\[{\rm{det (A}} - {\rm{I)}}\].

Note:
Matrix is the set of numbers arranged in the form of a rectangular array with some rows and columns. A square matrix is a matrix in which the number of rows equals the number of columns. The order of a matrix is the number of rows or columns of that matrix. Transpose of a matrix is a property of the matrix in which the rows are interchanged with columns and columns are interchanged with rows is known as the transpose of a matrix. The identity matrix is the matrix whose value of diagonal elements is 1 and the value of the rest elements is 0.