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How to find the determinant of the given elementary matrix by inspection? First row (1 0 0 0), second row (0 1 0 0), third row (0 0 -5 0), fourth row (0 0 0 1).

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Last updated date: 11th Jun 2024
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Answer
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Hint:
Here, we will firstly write the matrix given. Then we will use the theorem of the determinant of the triangular matrix to get the determinant of the given matrix which will be equal to the product of the diagonal elements of the matrix. The determinant of a matrix is a scalar value of the matrix which is computed from the elements of that matrix.

Complete step by step solution:
Given rows of the given matrix is first row (1 0 0 0), second row (0 1 0 0), third row (0 0 -5 0), fourth row (0 0 0 1).
First, we will write the given matrix. Let the given matrix be A, we get
\[A = \left[ {\begin{array}{*{20}{l}}
1&0&0&0\\
0&1&0&0\\
0&0&{ - 5}&0\\
0&0&0&1
\end{array}} \right]\]
Now we will find the determinant of the matrix.
Determinant is only applied to the square matrices and square matrix is the matrix which has number of rows of the matrix equal to the number of the column of the matrix. Determinant of a matrix is denoted as \[\left| A \right|\].
Now according to the theorem which states that the determinant of the triangular matrix is equal to the product of the diagonal elements. Therefore, by using this we will get the determinant of the given matrix, we get
\[ \Rightarrow \left| A \right| = \left| {\begin{array}{*{20}{l}}
1&0&0&0\\
0&1&0&0\\
0&0&{ - 5}&0\\
0&0&0&1
\end{array}} \right| = 1 \times 1 \times \left( { - 5} \right) \times 1\]
Now we will simply multiply to get the value of the determinant of the matrix. Therefore, we get
\[ \Rightarrow \left| A \right| = \left| {\begin{array}{*{20}{l}}
1&0&0&0\\
0&1&0&0\\
0&0&{ - 5}&0\\
0&0&0&1
\end{array}} \right| = - 5\]

Hence the determinant of the given matrix is equal to \[ - 5\].

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.