Answer

Verified

415.8k+ views

**Hint:**Here, we have to find the determinant of the given matrix. We will convert the given matrix into a Row Echelon form by using elementary row operations. We will then use the Row echelon form of the matrix to find the determinant of the given matrix. The determinant of a matrix is a value obtained after crossing out a row and column by multiplying the determinant of a square matrix.

**Complete Step by Step Solution:**

We are given with a matrix \[\left( {\begin{array}{*{20}{l}}0&3&1\\1&1&2\\3&2&4\end{array}} \right)\].

Now, we will reduce the given matrix to row echelon form by using elementary row operations.

First, we will interchange the first row and the second row, so we get

\[ \Rightarrow \left( {\begin{array}{*{20}{l}}0&3&1\\1&1&2\\3&2&4\end{array}} \right) = \left( {\begin{array}{*{20}{l}}1&1&2\\0&3&1\\3&2&4\end{array}} \right)\]

Now, we will transform the first element of the third row as\[1\] by using the operation \[{R_3} \to {R_3} - 3{R_1}\]. So, we get

\[ \Rightarrow \left( {\begin{array}{*{20}{l}}0&3&1\\1&1&2\\3&2&4\end{array}} \right) = \left( {\begin{array}{*{20}{l}}1&1&2\\0&3&1\\0&{ - 1}&{ - 2}\end{array}} \right)\]

Now, we will transform the second element of the third row as \[0\] by using the operation \[{R_3} \to {R_3} + \dfrac{{{R_2}}}{3}\]. So, we get

\[ \Rightarrow \left( {\begin{array}{*{20}{l}}0&3&1\\1&1&2\\3&2&4\end{array}} \right) = \left( {\begin{array}{*{20}{l}}1&1&2\\0&3&1\\0&0&{ - \dfrac{5}{3}}\end{array}} \right)\]

We will now find the determinant of the above matrix which is in row-echelon form.

\[ \Rightarrow \left| {\begin{array}{*{20}{l}}1&1&2\\0&3&1\\0&0&{ - \dfrac{5}{3}}\end{array}} \right| = 1\left| {\begin{array}{*{20}{l}}3&1\\0&{ - \dfrac{5}{3}}\end{array}} \right| - 1\left| {\begin{array}{*{20}{l}}0&1\\0&{ - \dfrac{5}{3}}\end{array}} \right| + 2\left| {\begin{array}{*{20}{l}}0&3\\0&0\end{array}} \right|\]

Simplifying the determinant, we get

\[ \Rightarrow \left| {\begin{array}{*{20}{l}}1&1&2\\0&3&1\\0&0&{ - \dfrac{5}{3}}\end{array}} \right| = 1\left( 3 \right)\left( { - \dfrac{5}{3}} \right) - 1\left( {0 - 0} \right) + 2\left( {0 - 0} \right)\]

\[ \Rightarrow \left| {\begin{array}{*{20}{l}}1&1&2\\0&3&1\\0&0&{ - \dfrac{5}{3}}\end{array}} \right| = 1 \times \left( 3 \right) \times \left( { - \dfrac{5}{3}} \right)\]

Multiplying the terms, we get

\[ \Rightarrow \left| {\begin{array}{*{20}{l}}1&1&2\\0&3&1\\0&0&{ - \dfrac{5}{3}}\end{array}} \right| = - 5\]

Since a row has been interchanged, then the final determinant has to be multiplied by \[\left( { - 1} \right)\] . Therefore, we get

\[ \Rightarrow \left| {\begin{array}{*{20}{l}}1&1&2\\0&3&1\\0&0&{ - \dfrac{5}{3}}\end{array}} \right| = \left( { - 1} \right) \times \left( { - 5} \right)\]

\[ \Rightarrow \left| {\begin{array}{*{20}{l}}1&1&2\\0&3&1\\0&0&{ - \dfrac{5}{3}}\end{array}} \right| = 5\]

**Therefore the value of the determinant of the row echelon form of the given matrix is \[5\].**

**Note:**

We know that for every square matrix, we can associate a number which is called as the determinant of the matrix. Row echelon form is any matrix that has the first non-zero element in the first row should be one and the elements below the main diagonal should be zero. Row echelon form of a matrix is also an upper triangular matrix. Whenever a row or a column is interchanged then the determinant has to be multiplied by a negative sign.

Recently Updated Pages

what is the correct chronological order of the following class 10 social science CBSE

Which of the following was not the actual cause for class 10 social science CBSE

Which of the following statements is not correct A class 10 social science CBSE

Which of the following leaders was not present in the class 10 social science CBSE

Garampani Sanctuary is located at A Diphu Assam B Gangtok class 10 social science CBSE

Which one of the following places is not covered by class 10 social science CBSE

Trending doubts

Which are the Top 10 Largest Countries of the World?

How do you graph the function fx 4x class 9 maths CBSE

The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths

Fill the blanks with the suitable prepositions 1 The class 9 english CBSE

Difference Between Plant Cell and Animal Cell

Why is there a time difference of about 5 hours between class 10 social science CBSE

Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE

Give 10 examples for herbs , shrubs , climbers , creepers

Draw a labelled sketch of the human eye class 12 physics CBSE