Questions: Invertible Matrices and Matrix Inverses

Question 1

A 3×3 matrix has a determinant of 0. What can we conclude about its inverse?

• A. It has no inverse because the determinant formula doesn't apply to singular matrices
• B. It has no inverse because the matrix collapses 3D space to a lower dimension, making recovery impossible
• C. It has an inverse, but computing it requires special numerical methods
• D. It is not a square matrix, so the inverse is undefined

Answer: (B)

A determinant of zero means the transformation squashes space down to a lower dimension — for example, collapsing a plane to a line. Multiple input vectors map to the same output, so there is no way to determine which input produced any given output. The inverse, by definition, must undo the transformation — but information lost in the collapse cannot be recovered. Options A and C are wrong because singularity (det = 0) is precisely what makes the inverse undefined, not a computational difficulty.

Question 2

Which property is necessary but NOT sufficient for a square matrix to be invertible?

• A. Being a square matrix
• B. Having a non-zero determinant
• C. Having linearly independent rows
• D. Having full rank

Answer: (A)

Squareness is required — a non-square matrix cannot have a two-sided inverse because it maps between spaces of different dimensions. But squareness alone is not enough: a square matrix can still have determinant zero, fail to have full rank, or have linearly dependent rows, all of which make it non-invertible. Options B, C, and D are each sufficient (as well as necessary) for invertibility — any one of them implies all the others.

Question 3

If a square matrix A satisfies AA⁻¹ = I, it automatically also satisfies A⁻¹A = I.

• T. True
• F. False

Answer: True

For square matrices, a one-sided inverse is automatically a two-sided inverse. This is a theorem in linear algebra: if A and B are square and AB = I, then BA = I as well. This is not true for non-square matrices, where left inverses and right inverses can exist independently — another reason invertibility requires squareness.

Question 4

A 2×3 matrix of full rank can be inverted using the same formula as a square matrix.

• T. True
• F. False

Answer: False

A 2×3 matrix maps ℝ³ into ℝ² — it compresses a three-dimensional space into two dimensions, necessarily discarding information. No matter how you construct a 'reverse' transformation, you cannot recover the original three-dimensional vector from two-dimensional output. True (two-sided) inverses only exist for square matrices. While left-inverses or right-inverses can exist for non-square matrices, they are not inverses in the full sense and the standard inverse formula does not apply.

Question 5

Why does a matrix with determinant zero have no inverse, even if it is square?

Model answer: A zero determinant means the matrix collapses space to a lower dimension — for example, mapping a plane onto a single line. When this happens, multiple distinct input vectors are sent to the same output vector. An inverse would need to reverse this mapping, but it cannot determine which of the many possible inputs produced any given output. Because information is irrecoverably lost in the collapse, no inverse exists.

The inverse must satisfy A⁻¹(Ax) = x for every input x. But if two different vectors x₁ and x₂ both satisfy Ax₁ = Ax₂ = b (which happens when the transformation is rank-deficient), then A⁻¹b cannot equal both x₁ and x₂ — a function must have a unique output. The determinant measuring zero area/volume is the geometric signature of this collapse.