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Q1. Calculate the determinant of H = [[1, 2, 1], [0, 1, 2], [1, 0, 1]]. (2020)
Solution:
Det(H) = 1(1*1 - 2*0) - 2(0*1 - 2*1) + 1(0*0 - 1*1) = 1(1) - 2(-2) + 1(-1) = 1 + 4 - 1 = 4.
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Q2. What is the value of det(H) for H = [[3, 2], [1, 5]]? (2021)
Solution:
Determinant of H = (3*5) - (2*1) = 15 - 2 = 13.
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Q3. What is the determinant of the matrix \( F = \begin{pmatrix} 1 & 2 & 3 \\ 0 & 1 & 4 \\ 5 & 6 & 0 \end{pmatrix} \)? (2022)
Solution:
Using the determinant formula for 3x3 matrices, we find \( 1(1*0 - 4*6) - 2(0*0 - 4*5) + 3(0*6 - 1*5) = 1(0 - 24) - 2(0 - 20) + 3(0 - 5) = -24 + 40 - 15 = 1 \).
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Q4. If F = [[2, 0], [0, 3]], what is det(F)? (2020)
Solution:
The determinant of F is calculated as (2*3) - (0*0) = 6.
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Q5. If G = [[1, 1], [1, -1]], find det(G). (2022)
Solution:
Determinant of G = (1*-1) - (1*1) = -1 - 1 = -2.
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Q6. Find the determinant of E = [[4, 2], [1, 3]]. (2023)
Solution:
Det(E) = (4*3) - (2*1) = 12 - 2 = 10.
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Q7. Calculate the determinant of D = [[3, 2, 1], [1, 0, 2], [0, 1, 3]]. (2023)
Solution:
Det(D) = 3(0*3 - 2*1) - 2(1*3 - 0*2) + 1(1*1 - 0*0) = 3(0 - 2) - 2(3) + 1(1) = -6 - 6 + 1 = -11.
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Q8. Calculate the determinant of D = [[4, 2], [3, 1]]. (2020)
Solution:
Determinant of D = (4*1) - (2*3) = 4 - 6 = -2.
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Q9. What is the determinant of G = [[5, 4], [2, 3]]? (2020)
Solution:
Det(G) = (5*3) - (4*2) = 15 - 8 = 7.
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Q10. What is the determinant of the matrix \( H = \begin{pmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{pmatrix} \)? (2023)
Solution:
The determinant of this matrix is 0 because the rows are linearly dependent.
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