Question:
If
\[
A=
\begin{bmatrix}
1&2\\
3&4
\end{bmatrix},
\]
then \(\det(A)\) is equal to:
If
\[
A=
\begin{bmatrix}
1&2\\
3&4
\end{bmatrix},
\]
then \(\det(A)\) is equal to:
Show Hint
For a \(2\times2\) matrix,
\[
\det
\begin{bmatrix}
a&b\\
c&d
\end{bmatrix}
=
ad-bc.
\]
Always multiply the main diagonal first and then subtract the product of the other diagonal.
Updated On: Jun 10, 2026
- \(-2\)
- \(-1\)
- \(1\)
- \(2\)
Show Solution
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The Correct Option is A
Solution and Explanation
Concept:
The determinant of a square matrix is a numerical value associated with the matrix. Determinants are extremely useful in solving systems of equations, finding inverses of matrices, and studying linear transformations.
For a \(2\times2\) matrix
\[
\begin{bmatrix}
a&b\\
c&d
\end{bmatrix},
\]
the determinant is calculated using
\[
ad-bc.
\]
This is one of the most fundamental formulas in matrix algebra.
Step 1: Identify the entries of the matrix. From \[ A= \begin{bmatrix} 1&2\\ 3&4 \end{bmatrix}, \] we have \[ a=1,\quad b=2,\quad c=3,\quad d=4. \]
Step 2: Write the determinant formula. \[ \det(A)=ad-bc. \]
Step 3: Substitute the values. \[ \det(A) = (1)(4)-(2)(3). \]
Step 4: Perform the calculations. \[ = 4-6. \] \[ = -2. \]
Step 5: Verification. The determinant calculation follows the standard rule exactly and therefore the result is correct.
Step 6: Final Conclusion. \[ \boxed{\det(A)=-2} \] Hence the correct answer is \[ \boxed{\text{Option (A)}}. \]
Step 1: Identify the entries of the matrix. From \[ A= \begin{bmatrix} 1&2\\ 3&4 \end{bmatrix}, \] we have \[ a=1,\quad b=2,\quad c=3,\quad d=4. \]
Step 2: Write the determinant formula. \[ \det(A)=ad-bc. \]
Step 3: Substitute the values. \[ \det(A) = (1)(4)-(2)(3). \]
Step 4: Perform the calculations. \[ = 4-6. \] \[ = -2. \]
Step 5: Verification. The determinant calculation follows the standard rule exactly and therefore the result is correct.
Step 6: Final Conclusion. \[ \boxed{\det(A)=-2} \] Hence the correct answer is \[ \boxed{\text{Option (A)}}. \]
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