Question

If $$ A = \begin{pmatrix} 2 & 3 \\ 1 & k \end{pmatrix} $$ and $\det(A) = 7$, find the value of $ k $.

• A. \(1\)
• B. \(2\)
• C. \(5\)
• D. \(4\)

Hint:

Tip: Carefully apply determinant formula and check arithmetic to avoid mistakes.

Answer 1

The Correct Option is C

The determinant of matrix \(A\) is given by the formula: 

\( \det(A) = ad - bc \)

For the matrix \( A = \begin{pmatrix} 2 & 3 \\ 1 & k \end{pmatrix} \), \( a = 2 \), \( b = 3 \), \( c = 1 \), and \( d = k \).

Thus, the determinant equation becomes:

\( \det(A) = (2)(k) - (3)(1) = 2k - 3 \)

Given that the determinant \(\det(A) = 7\), we have the equation:

\( 2k - 3 = 7 \)

Solving for \( k \), add 3 to both sides:

\( 2k = 10 \)

Next, divide by 2:

\( k = 5 \)

Therefore, the value of \( k \) is 5.

Answer 2

Step 1: Recall determinant formula 
\[ \det(A) = ad - bc \] For \[ A = \begin{pmatrix} 2 & 3 \\ 1 & k \end{pmatrix}, \] we have \(a=2\), \(b=3\), \(c=1\), and \(d=k\).

Step 2: Write determinant equation 
\[ \det(A) = 2 \times k - 3 \times 1 = 2k - 3 \]

Step 3: Use given determinant value 
\[ 2k - 3 = 7 \implies 2k = 10 \implies k = 5 \]