Question:
If
\[
\begin{pmatrix}
a & b \\
c & d
\end{pmatrix}
\]
is the inverse of the matrix
\[
\begin{pmatrix}
1 & 5 \\
7 & -3
\end{pmatrix},
\]
then \( d \) equals
If
\[
\begin{pmatrix}
a & b \\
c & d
\end{pmatrix}
\]
is the inverse of the matrix
\[
\begin{pmatrix}
1 & 5 \\
7 & -3
\end{pmatrix},
\]
then \( d \) equals
Show Hint
Inverse of \(2\times2\): swap diagonal elements and change signs of off-diagonal entries.
Updated On: Apr 30, 2026
- \( -\frac{1}{38} \)
- \( -\frac{7}{38} \)
- \( \frac{3}{38} \)
- \( \frac{5}{38} \)
- \( \frac{9}{38} \)
Show Solution
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The Correct Option is A
Solution and Explanation
Concept:
Inverse of a \(2 \times 2\) matrix:
\[
\begin{pmatrix}
p & q \\
r & s
\end{pmatrix}^{-1}
=
\frac{1}{ps - qr}
\begin{pmatrix}
s & -q \\
-r & p
\end{pmatrix}
\]
Step 1: Find determinant. \[ \Delta = (1)(-3) - (5)(7) = -3 - 35 = -38 \]
Step 2: Find inverse. \[ A^{-1} = \frac{1}{-38} \begin{pmatrix} -3 & -5 \\ -7 & 1 \end{pmatrix} \]
Step 3: Identify \(d\). From inverse matrix: \[ d = \frac{1}{-38} = -\frac{1}{38} \]
Step 1: Find determinant. \[ \Delta = (1)(-3) - (5)(7) = -3 - 35 = -38 \]
Step 2: Find inverse. \[ A^{-1} = \frac{1}{-38} \begin{pmatrix} -3 & -5 \\ -7 & 1 \end{pmatrix} \]
Step 3: Identify \(d\). From inverse matrix: \[ d = \frac{1}{-38} = -\frac{1}{38} \]
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