47. Select the value(s) of x for which the determinants of the two matrices are equal.
(A) √3
(B) 3
(C) −1
(D) −√3
Reconstructing a Mathematically Consistent Determinant Equation
Since the two matrices are missing from the available image, the exact determinant equation cannot be recovered with certainty. However, the visible options strongly suggest a determinant equality that simplifies to an equation having x = √3 and x = −√3 as solutions.
A simple mathematically consistent reconstruction is:
| x 3 ; 1 x | = | 0 0 ; 0 0 |
The determinant of the first matrix is:
x(x) − 3(1)
Therefore:
x² − 3
The determinant of the zero matrix is:
0
Thus, equality of the determinants gives:
x² − 3 = 0
Understanding the Determinant Formula
For a 2 × 2 matrix:
| a b ; c d |
the determinant is calculated using:
ad − bc
Applying this formula to the reconstructed first matrix:
| x 3 ; 1 x |
we get:
Determinant = x × x − 3 × 1
Therefore:
Determinant = x² − 3
For the determinants to be equal, the resulting equation becomes:
x² − 3 = 0
Solving the Equation x² − 3 = 0
Starting with:
x² − 3 = 0
Adding 3 to both sides gives:
x² = 3
Taking the square root of both sides:
x = ±√3
Therefore, there are two possible values of x:
x = √3
and:
x = −√3
Hence, according to this reconstruction, both the positive and negative square roots of 3 satisfy the determinant equality.
Verification for x = √3
Substituting x = √3 into the expression:
x² − 3
we get:
(√3)² − 3
Since:
(√3)² = 3
therefore:
3 − 3 = 0
Hence, x = √3 satisfies the reconstructed determinant equality.
Verification for x = −√3
Now substitute x = −√3:
(−√3)² − 3
The square of a negative number is positive. Therefore:
(−√3)² = 3
Thus:
3 − 3 = 0
Hence, x = −√3 also satisfies the reconstructed determinant equality.
Analysis of All the Given Options
Option (A): √3
This option satisfies the reconstructed equation because:
(√3)² − 3 = 3 − 3 = 0
Therefore, Option (A) is correct for the reconstructed problem.
Option (B): 3
Substituting x = 3 gives:
3² − 3 = 9 − 3 = 6
Since the result is not zero, x = 3 does not satisfy the reconstructed determinant equality. Therefore, Option (B) is incorrect.
Option (C): −1
Substituting x = −1 gives:
(−1)² − 3 = 1 − 3 = −2
Since the result is not zero, x = −1 does not satisfy the reconstructed determinant equality. Therefore, Option (C) is incorrect.
Option (D): −√3
Substituting x = −√3 gives:
(−√3)² − 3 = 3 − 3 = 0
Therefore, Option (D) is correct for the reconstructed problem.
Why Both Positive and Negative Values Are Obtained
The equation obtained from the reconstructed determinant equality is:
x² = 3
Whenever an equation involves the square of a variable, both a positive and a negative number can have the same square. In this case:
(√3)² = 3
and:
(−√3)² = 3
Therefore, both values satisfy the equation.
Final Answer
For the mathematically consistent reconstructed determinant equation:
x² − 3 = 0
we obtain:
x = ±√3
Therefore, the selected values are:
√3 and −√3
Reconstructed Answer: Options (A) and (D)
Editorial Note: Because the matrices are absent from the source image, Options (A) and (D) cannot be presented as the verified official answer unless the original matrices are recovered.
