Determinants – important questions for Boards 2027
CBSE Class 12 Maths
Evaluate |[[1, 0, 0], [2, 3, 0], [4, 5, 6]]| by expanding along the first row.
Answer
|1 0 0| |2 3 0| |4 5 6| = 1(3 × 6 − 0 × 5) − 0(2 × 6 − 0 × 4) + 0(2 × 5 − 3 × 4) = 18 ∴ 18
If A = 2B for 3 × 3 matrices and |B| = 5, calculate |A|.
Answer
|A| = 2³|B| = 8 × 5 ∴ |A| = 40
Find the determinants of A=[1 2; 2 4] and B=[1 2; 3 4], and classify each matrix as singular or nonsingular.
Answer
A square matrix is singular if its determinant is zero and nonsingular if its determinant is non-zero. |A| = |1 2 2 4| = 1 × 4 − 2 × 2 = 0 ∴ |A| = 0; A is singular. |B| = |1 2 3 4| = 1 × 4 − 2 × 3 = −2 ∴ |B| = −2; B is nonsingular.
Find the area of the triangle with vertices (1, 0), (6, 0) and (4, 3), using the determinant of their coordinates.
Answer
Area = 1/2 × |determinant of the coordinates| Area = 1/2 × | 1 0 1 6 0 1 4 3 1 | = 1/2 × |15| ∴ Area = 15/2 square units
Find the adjoint of A=[2 −3; 4 1], showing how the transpose of its cofactor matrix gives the result.
Answer
The adjoint of a square matrix is the transpose of its matrix of cofactors. A = [ 2 −3 4 1 ] [Aᵢⱼ] = [ (−1)¹⁺¹(1) (−1)¹⁺²(4) (−1)²⁺¹(−3) (−1)²⁺²(2) ] = [ 1 −4 3 2 ] adj A = [Aᵢⱼ]ᵀ = [ 1 3 −4 2 ] ∴ adj A = [ 1 3 −4 2 ]
For A=[1 2; 2 4] and B=[3; 7], find (adj A)B and determine whether AX=B has a solution.
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Consider the two systems 0x+0y=0, 0x+0y=0 and 0x+0y=1, 0x+0y=0. For each, calculate (adj A)B and determine whether the system is consistent, showing what the zero product alone does not decide.
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Evaluate |[[2, 1, 0], [3, 0, 0], [1, 4, 5]]| by expanding along the row or column containing the most zeros.
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For the entry in row 2, column 3 of a determinant, calculate its cofactor sign (−1)^(i+j).
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A triangle has area 4 square units. Which values can the determinant formed from its vertex coordinates take? (A) 4 only (B) −4 only (C) 8 or −8 (D) 0
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42 more probable questions in this chapter, every one with a full answer.
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