Determinants — Practice

Question 1

Evaluate |[[3,-1],[2,4]]|.

Accepted Answer

3*4 - (-1)*2 = 12+2 = 14. Answer: |A| = 14.

Question 2

Is A=[[2,4],[1,2]] singular or non-singular?

Accepted Answer

|A| = 2*2-4*1 = 4-4 = 0. Answer: A is singular (and therefore not invertible).

Question 3

Find the area of the triangle with vertices (0,0), (4,0), (0,3).

Accepted Answer

Delta = (1/2)|det[[0,0,1],[4,0,1],[0,3,1]]| = (1/2)|0(0-3)-0(4-0)+1(12-0)| = (1/2)(12) = 6. Answer: Area = 6 square units.

Question 4

Find the cofactor A21 of the matrix [[1,2,3],[4,5,6],[7,8,10]].

Accepted Answer

M21 = det[[2,3],[8,10]] = 20-24 = -4. A21 = (-1)^(2+1) * M21 = -(-4) = 4. Answer: A21 = 4.

Question 5

Find the inverse of A=[[3,1],[2,1]].

Accepted Answer

|A| = 3*1-1*2 = 1. adj(A) = [[1,-1],[-2,3]]. A^{-1} = (1/1)*[[1,-1],[-2,3]]. Answer: A^{-1} = [[1,-1],[-2,3]].

Question 6

Solve using the matrix method: 3x+y=7, x+2y=4.

Accepted Answer

A=[[3,1],[1,2]], B=[7,4]'. |A|=6-1=5. adj(A)=[[2,-1],[-1,3]]. A^{-1}=(1/5)[[2,-1],[-1,3]]. X=A^{-1}B=(1/5)[2*7-1*4, -1*7+3*4]=(1/5)[10,5]=[2,1]. Answer: x=2, y=1.

Question 7

Examine the consistency of x+2y=3, 2x+4y=6.

Accepted Answer

A=[[1,2],[2,4]]. |A|=4-4=0, so A is singular. Checking: the second equation is exactly 2 times the first, so the equations represent the same line. Answer: The system is consistent, with infinitely many solutions (every point on the line x+2y=3).

Question 8

If |A|=5 for a 3x3 matrix A, find |adj(A)| and |2A|.

Accepted Answer

For an n x n matrix, |adj(A)|=|A|^(n-1); here n=3, so |adj(A)|=5^2=25. Also |kA|=k^n|A|; here |2A|=2^3 * 5=40. Answer: |adj(A)|=25, |2A|=40.

Practice — Determinants

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