Matrices & Determinants

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Q1. Calculate the determinant of G = [[1, 1, 1], [1, 2, 3], [1, 3, 6]]. (2022)

Solution:

The determinant of G is 0 because the rows are linearly dependent.

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Q2. If H = [[1, 2], [2, 4]], what is det(H)? (2020)

Solution:

The determinant of H is (1*4) - (2*2) = 4 - 4 = 0.

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Q3. Calculate the determinant of D = [[4, 2], [1, 3]]. (2020)

Solution:

Determinant of D = (4*3) - (2*1) = 12 - 2 = 10.

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Q4. What is the determinant of the matrix G = [[2, 4], [1, 3]]? (2021)

Solution:

The determinant of G is (2*3) - (4*1) = 6 - 4 = 2.

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Q5. Find the determinant of E = [[1, 2, 3], [4, 5, 6], [7, 8, 9]]. (2019)

Solution:

Determinant of E = 0 (rows are linearly dependent).

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Q6. Find the determinant of the matrix D = [[4, 2], [3, 1]]. (2023)

Solution:

The determinant of D is calculated as (4*1) - (2*3) = 4 - 6 = -2.

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Q7. What is the determinant of the matrix \( H = \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix} \)? (2022)

Solution:

The determinant of the identity matrix is always 1.

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Q8. Calculate the determinant of \( D = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix} \). (2021)

Solution:

The determinant is \( 0*0 - 1*1 = 0 - 1 = -1 \).

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Q9. If I = [[1, 2], [2, 4]], what is det(I)? (2021)

Solution:

The determinant of I is 0 because the rows are linearly dependent.

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Q10. If I = [[1, 0, 2], [0, 1, 3], [1, 0, 4]], find det(I). (2021)

Solution:

Using cofactor expansion, det(I) = 1(1*4 - 3*0) - 0 + 2(0*0 - 1*1) = 4 - 2 = 2.