edited by
20,633 views
29 votes
29 votes

The determinant of the matrix given below is

$$\begin{bmatrix}
0 &1  &0  &2 \\
 -1&  1&  1& 3\\
 0&0  &0  & 1\\
 1&  -2&  0& 1
\end{bmatrix}$$

  1. $-1$
  2. $0$
  3. $1$
  4. $2$
edited by

8 Answers

Best answer
28 votes
28 votes

$$\begin{bmatrix}
0 &1  &0  &2 \\
 -1&  1&  1& 3\\
 0&0  &0  & 1\\
 1&  -2&  0& 1
\end{bmatrix}$$
Reduce this matrix into Upper Triangular matrix using row and column transformations:
$R1\leftrightarrow R2$
$R3\leftrightarrow R4$

$$\begin{bmatrix}
 -1&  1&  1& 3\\
0 &1  &0  &2 \\
1&  -2&  0& 1\\
 0&0  &0  & 1
\end{bmatrix}$$
$R3\leftarrow R1 + R3$
$R3\leftarrow R2 - R3$

Resulting Upper Triangular matrix will be:

$$\begin{bmatrix}
 -1&  1&  1& 3\\
0 &1  &0  &2 \\
0&  0&  1& 2\\
 0&0  &0  & 1
\end{bmatrix}$$
Determinant will be product of diagonal elements = $-1*1*1*1=-1$

Hence, A is correct option!

selected by
31 votes
31 votes
$+0\begin{bmatrix} 1 & 1 & 3\\ 0 & 0 & 1\\ -2 & 0 & 1 \end{bmatrix}$$-1\begin{bmatrix} -1 & 1 & 3\\ 0 & 0 & 1\\ 1 & 0 & 1 \end{bmatrix}$$+0\begin{bmatrix} -1 & 1 & 3\\ 0 & 0 & 1\\ 1 & -2 & 1 \end{bmatrix}$$-2\begin{bmatrix} -1 & 1 & 1\\ 0 & 0 & 0\\ 1 & -2 & 0 \end{bmatrix}$

=$-1\begin{bmatrix} -1 & 1 & 3\\ 0 & 0 & 1\\ 1 & 0 & 1 \end{bmatrix}$$=-1$
2 votes
2 votes

$$\begin{bmatrix}
0 &1  &0  &2 \\
 -1&  1&  1& 3\\
 0&0  &0  & 1\\
 1&  -2&  0& 1
\end{bmatrix}$$

do $R_1\leftrightarrow R_3$ and then expand $4*4$ matrix along $R_1$  and at next step expand $3*3$ matrix along $C_3$

 

$=-\begin{vmatrix}
 {\color{Blue} 0}&   {\color{Blue} 0}&   {\color{Blue} 0}&  {\color{Blue} 1}\\
-1 &1  &1  &3 \\
0&  1&  0& 2\\
 1&-2  &0  & 1
\end{vmatrix} =- {\color{Blue} 1}\begin{vmatrix} -1 & 1&  {\color{Red} 1}\\ 0& 1& {\color{Red} 0}\\ 1 & -2 & {\color{Red} 0} \end{vmatrix}= -{\color{Blue} 1}* {\color{Red}{ -1}} \begin{vmatrix} 0 &1 \\ 1 & -2 \end{vmatrix}= -{\color{Blue} 1}* {\color{Red}{ -1}}-1=-1$

 

Hence, A is correct option!

Answer:

Related questions

18 votes
18 votes
3 answers
1
Kathleen asked Sep 29, 2014
3,841 views
The determinant of the matrix $\begin{bmatrix} 6 & -8 & 1 & 1 \\ 0 & 2 & 4 & 6 \\ 0 & 0 & 4 & 8 \\ 0 & 0 & 0 & -1 \end{bmatrix}$$11$$-48$$0$$-24$
38 votes
38 votes
6 answers
2
Arjun asked Sep 23, 2014
9,194 views
Which one of the following does NOT equal $$\begin{vmatrix} 1 & x & x^{2}\\ 1& y & y^{2}\\ 1 & z & z^{2} \end{vmatrix} \quad ?$$$\begin{vmatrix} 1& x(x+1)& x+1\\ 1& y(y+1...
18 votes
18 votes
3 answers
3
0 votes
0 votes
1 answer
4
admin asked Apr 1, 2020
578 views
$M$ is a square matrix of order $’n’$ and its determinant value is $5.$ If all the elements of $M$ are multiplied by $2,$ its determinant value becomes $40.$ The valu...