Question

For what value of 'K' the matrix [k234]has no inverse.

Answer 1

$\underline{\underline{\Huge{\textbf{Question:}}}}$

$\mathsf{For\: what\: value\: of\: `K`\: the\: matrix}\\\mathsf{\left|\begin{array}{cc}k&2\\3&4\end{array}\right|\; has\: no\: Inverse}$




$\underline{\Huge{\textbf{Concept Behind:}}}$

$\bullet\: \sf\textsf{Only Non-singular matrix can have inverse}$

$\bullet\: \sf\textsf{If there exists no inverse for a matrix, then}\\\sf\textsf{it is called \textbf{Singular Matrix or Non Invertible Matrix}}$

$\sf\textsf{Consider general form of a 2 \times 2 Matrix}$

$\mathsf{Say\: A = \left[\begin{array}{cc}a&b\\c&d\end{array}\right]}$

$\sf\textsf{Inverse of the Matrix A is given by,}$

$\qquad \LARGE{\boxed{\bigstar \; \; \mathbf{A^{-1} = \dfrac{1}{|A|}\: \left[\begin{array}{cc}d&-b\\-c&a\end{array}\right]}}}$

$\sf\textsf{Where,}$

$\qquad \LARGE{\boxed{\bigstar \; \; \mathbf{|A| =} \textbf{(ad-bc)}}}$ ———[1]

$\maltese\: \sf\textsf{If the determinant of A (det A) is equal to zero,}\\\sf\textsf{then the Result tends to infinity and}\\\sf\textsf{the Matrix Inversion cannot be possible}$

$\therefore \sf\textsf{The matrix inversion can only be possible if}\\\sf\textbf{the determinant of the matrix is not zero}$




$\underline{\underline{\Huge{\textbf{Solution:}}}}$

$\underline{\LARGE{\textbf{Step-1:}}}$

$\sf\textsf{Find the Determinant of Given Matrix}$

Let the Given Matrix be A.

$\implies \mathsf{A = \left|\begin{array}{cc}k&2\\3&4\end{array}\right|}$

$\sf\textsf{Comparing it with general form of 2x2 Matrix}\\\sf\textsf{We get,}\\\quad\\\textbf{a= k; b = 2; c = 3; d = 4}$

$\sf\textsf{On Substituting in [1]}$

$\implies \mathsf{|A| = 4(k) - 2(3)}$

$\implies \mathsf{|A| = 4(k) - 6}$




$\underline{\LARGE{\textbf{Step-2:}}}$

$\sf\textsf{Equate the result of determinant to zero}\\\sf\textsf{to obtain value of k}$

$\implies \mathsf{|A| = 0}$

$\implies \mathsf{4(k) - 6 = 0}$

$\implies \mathsf{4(k) = 6}$

$\implies \mathsf{2k = 3}$

$\implies \mathsf{k = \dfrac{3}{2}}$





$\therefore$

$\blacksquare\; \, \mathsf{The value\: of\: `K`\: for which the\: matrix}\\\mathsf{\left|\begin{array}{cc}k&2\\3&4\end{array}\right|\; has\: no\: Inverse} = \Large{\mathbf{\dfrac{3}{2}}}$

Answer 2

Hi

Nice question

The answer for the question is k=3/2