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Question: If A is a square matrix of order 3 and \[{{\rm{A}}^{\rm{T}}}\] denotes the transpose of matrix A, \[...

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

Explanation

Solution

Here, in this question, we have to use the basic concept of the matrix to find the value ofdet(A−I){\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 ofdet(A−I){\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. detA=1{\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 indet(A−I){\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)

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.