Question

The correct statement regarding the determinants (Det) of matrices R, S and T is 

 

• A. Det(R) = Det(S) ≠ Det(T)
• B. Det(R) = Det(T) ≠ Det(S)
• C. Det(R) = Det(S) = Det(T)
• D. Det(R), Det(S), Det(T) are all different

Hint:

When solving determinant problems, remember to carefully compute each determinant and compare the values.

Answer 1

The Correct Option is B

1. Calculating the Determinant of R

$$R = \begin{vmatrix} 3 & 2 & 4 \\ 4 & 5 & 7 \\ 1 & 3 & 8 \end{vmatrix}$$

We use the cofactor expansion along the first row:

$$\text{Det}(R) = 3 \begin{vmatrix} 5 & 7 \\ 3 & 8 \end{vmatrix} - 2 \begin{vmatrix} 4 & 7 \\ 1 & 8 \end{vmatrix} + 4 \begin{vmatrix} 4 & 5 \\ 1 & 3 \end{vmatrix}$$

$$\text{Det}(R) = 3((5)(8) - (7)(3)) - 2((4)(8) - (7)(1)) + 4((4)(3) - (5)(1))$$

$$\text{Det}(R) = 3(40 - 21) - 2(32 - 7) + 4(12 - 5)$$

$$\text{Det}(R) = 3(19) - 2(25) + 4(7)$$

$$\text{Det}(R) = 57 - 50 + 28$$

$$\text{Det}(R) = 7 + 28 = \mathbf{35}$$

2. Calculating the Determinant of S

$$S = \begin{vmatrix} 2 & 3 & 4 \\ 5 & 4 & 7 \\ 3 & 1 & 8 \end{vmatrix}$$

We use the cofactor expansion along the first row:

$$\text{Det}(S) = 2 \begin{vmatrix} 4 & 7 \\ 1 & 8 \end{vmatrix} - 3 \begin{vmatrix} 5 & 7 \\ 3 & 8 \end{vmatrix} + 4 \begin{vmatrix} 5 & 4 \\ 3 & 1 \end{vmatrix}$$

$$\text{Det}(S) = 2((4)(8) - (7)(1)) - 3((5)(8) - (7)(3)) + 4((5)(1) - (4)(3))$$

$$\text{Det}(S) = 2(32 - 7) - 3(40 - 21) + 4(5 - 12)$$

$$\text{Det}(S) = 2(25) - 3(19) + 4(-7)$$

$$\text{Det}(S) = 50 - 57 - 28$$

$$\text{Det}(S) = -7 - 28 = \mathbf{-35}$$

3. Calculating the Determinant of T

$$T = \begin{vmatrix} 3 & 4 & 1 \\ 2 & 5 & 3 \\ 4 & 7 & 8 \end{vmatrix}$$

We use the cofactor expansion along the first row:

$$\text{Det}(T) = 3 \begin{vmatrix} 5 & 3 \\ 7 & 8 \end{vmatrix} - 4 \begin{vmatrix} 2 & 3 \\ 4 & 8 \end{vmatrix} + 1 \begin{vmatrix} 2 & 5 \\ 4 & 7 \end{vmatrix}$$

$$\text{Det}(T) = 3((5)(8) - (3)(7)) - 4((2)(8) - (3)(4)) + 1((2)(7) - (5)(4))$$

$$\text{Det}(T) = 3(40 - 21) - 4(16 - 12) + 1(14 - 20)$$

$$\text{Det}(T) = 3(19) - 4(4) + 1(-6)$$

$$\text{Det}(T) = 57 - 16 - 6$$

$$\text{Det}(T) = 41 - 6 = \mathbf{35}$$

4. Conclusion

Comparing the results:

$\text{Det}(R) = 35$

$\text{Det}(S) = -35$

$\text{Det}(T) = 35$

Therefore, $\text{Det}(R) = \text{Det}(T)$ and both are not equal to $\text{Det}(S)$.

$$\mathbf{\text{Det}(R) = \text{Det}(T) \neq \text{Det}(S)}$$

This corresponds to Option 2.