Question:

If ∣1−1x1x1x−11∣\begin{vmatrix}1&-1&x\\1&x&1\\x&-1&1\end{vmatrix} has no inverse, then the real value of xx is

Updated On: Jan 30, 2025
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The Correct Option is D

Solution and Explanation

∣1−1x1x1x−11∣\begin{vmatrix}1 & -1 & x \\ 1 & x & 1 \\ x & -1 & 1\end{vmatrix} 
Has no inverse;
⇒∣A∣=0\Rightarrow|A|=0
x+1+1(1−x)+x(−1−x2)=0x+1+1(1-x)+x\left(-1-x^{2}\right)=0
x+1+1−x−x−x3=0x+1+1-x-x-x^{3}=0
x3−x−2=0x^{3}-x-2=0
⇒x3+2x−x−2=0\Rightarrow x^{3}+2 x-x-2=0
(x2−1)(x+2)=0\left(x^{2}-1\right)(x+2)=0
⇒x=±1,x=−2\Rightarrow x=\pm 1, x=-2 
Real value of x=1x=1

 
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Concepts Used:

Transpose of a Matrix

The matrix acquired by interchanging the rows and columns of the parent matrix is called the Transpose matrix. The transpose matrix is also defined as - “A Matrix which is formed by transposing all the rows of a given matrix into columns and vice-versa.”

The transpose matrix of A is represented by A’. It can be better understood by the given example:

A Matrix
A' Matrix
The transpose matrix of A is denoted by A’

Now, in Matrix A, the number of rows was 4 and the number of columns was 3 but, on taking the transpose of A we acquired A’ having 3 rows and 4 columns. Consequently, the vertical Matrix gets converted into Horizontal Matrix.

Hence, we can say if the matrix before transposing was a vertical matrix, it will be transposed to a horizontal matrix and vice-versa.

Read More: Transpose of a Matrix