Determinants — Questions & Answers

Sample questions from Determinants

1. Q1. For det([[x,3,7],[2,x,2],[7,6,x]]) = 0, the sum of all three roots in x is:

   • A.0✓
   • B.9
   • C.-9
   • D.67
   Solution

   Expanding gives the cubic x^3 - 67x + 126 = 0. The coefficient of x^2 is 0, so by Vieta the sum of the roots is 0.

2. Q2. The adjoint of a square matrix A is:

   • A.The transpose of the matrix of cofactors of A✓
   • B.The inverse of A
   • C.The determinant of A
   • D.The transpose of A
   Solution

   adj(A) = transpose of the cofactor matrix of A. It is used to find A⁻¹ = adj(A)/|A|.

3. Q3. If A is a square matrix of order 3 satisfying A·(adj A) = 125 I, then det(adj A) equals:

   • A.125
   • B.25
   • C.15625✓
   • D.625
   Solution

   A·adj A = det(A) I gives det(A) = 125. Then det(adj A) = [det A]^2 = 125^2 = 15625.

4. Q4. The determinant det([[x+y,y+z,z+x],[z,x,y],[1,1,1]]) equals:

   • A.x+y+z
   • B.2(x+y+z)
   • C.xyz
   • D.0✓
   Solution

   R1 -> R1 + R2 makes every entry of the first row equal to x+y+z, i.e. proportional to R3. Two proportional rows force the determinant to 0.

5. Q5. For the lower triangular matrix A = [[5,0,0],[-2,3,0],[7,1,-4]], det(A) equals:

   • A.12
   • B.60
   • C.-12
   • D.-60✓
   Solution

   Triangular matrix: det = 5 × 3 × (-4) = -60.

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