Let A be a 3 times 3 matrix and suppose we know that -2a_1 + 1 a_2 + 3a_3 = 0 wh

Question

Let A be a 3 times 3 matrix and suppose we know that -2a_1 + 1 a_2 + 3a_3 = 0 where a_1, a_2 and a_3 are the columns of A. Write a non-trivial solution to the system Ax = 0 x = Is A singular or nonsingular? Check the correct answer below. A. The matrix A is singular because it is a square matrix. B. The matrix A is nonsingular because the homogeneous systems Ax = 0 has a non-trivial solution. C. The matrix A is singular because the homogeneous systems Ax = 0 has a non-trivial solution. D. The matrix A is nonsingular because it is a square matrix.

Explanation / Answer

If A is non–singular, the homogeneous system AX = 0 has only the trivial solution because if A is non–singular and AX = 0, then X = A-10 = 0

Let x1,x2,..xn be the elements of X. Then AX= x1a1 + x2a2 + ....+ xn an . where ai is the ith column vector of A.

Here, we need to find out the non-trivial solution for AX=0

One obvious solution is X=0 and it is the trivial solution.

But, Given -2a1+1a2+3a3 = 0 ---->eq(1)

Also, A is a 3 X 3 matrix and let x1,x2,x3 be the elements of X and a1,a2,a3 be the column vectors of A.

Therefore, AX=0 => x1a1 + x2a2+x3a3=0----->eq(2)

Comparing eq(1) and eq(2), we get x1=-2, x2=1 and x3=3.

Therefore, other than X=0 which is trivial , there exists another solution which is non-trivial (i.e X not equal to 0)

Non-trivial solution is X= [ -2 1 3 ] (Note: X is a 3x1 matrix and -2 1 3 should be written in a single column) .

Therefore, there exists a non-trivial solution. => AX=0 has a non-trivial solution and hence A is singular.

In short, The system Ax=b of n linear equations has a unique solution if and only if A is nonsingular. Here,Since X can be the 0 vector and vector <-2,1,3>, the solution isn't unique and therefore, A must be singular.

Therefore A is singular because the homogeneous system AX=0 has a non-trivial solution

Hence, option 3 is the correct option

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