Solution

If A is a singular matrix then |A| = 0

Solution

Hence, the value of det(2A^-1BC) is 16.

Solution

Solution

Solution

Find the least squares solution to

Solution

Solution

Solution

x = 2,

xp = 2

∴ p = 1

z = 4

zp + y = 9

(4)(1) + y = 9

y = 9 - 4

∴ y = 5

lower diagonal entry I₂₂ = 5

Solution

Solution

Solution

T : R^3 → R^3 be the linear transformation defined by T(x, y, z) = (x + y, y + z, z + x) for all (x, y, z) ∈ R^3

Solution

Note that the range R(A) is the column space of A. Thus,

R(A) = Span{A₁,A₂,A₃,A₄},

where Ai is the i-th column vector of A.

We reduce the matrix A by elementary row operations as follows.

We have

The first and the third column contain the leading 1’s.

Thus, the set {A1,A3} is a basis of the range R(A), which consists of the first and the third column vectors of A

we know that {A1,A3}is a basis of the range of A

Hence the dimension of the range is 2

Thus the rank of A, which is the dimension of the range R(A), is 2

Recall the rank-nullity theorem. Since A is a 3×4 matrix, we have

rank of A+nullity of A= 4.

Since we know that the rank of A is 2, it follows from the rank-nullity theorem that the nullity of A is 2

Solution

For any matrix, singular values and eigenvalues holds a relation given by

Hence, singular values for a matrix are always non-negative.

In this case, √4=2, √9=3. Hence, 2 and 3 can be the singular values.

Solution

We can try all options. Here, let’s try option D.

Clearly, it is giving matrix A. Hence, D is the correct answer.

Solution

The question is asking for a non-zero singular value. We know for an SVD for a real matrix the singular value cannot be complex or negative. So, the only answer we are left with is the positive real number.