Key Concepts & Formulas
- $$\det(AB) = \det(A) \cdot \det(B)$$
- $$\text{rank}(A) + \text{nullity}(A) = n$$
- $$A\mathbf{v} = \lambda \mathbf{v}$$
- $$\det(A - \lambda I) = 0$$
- Gram-Schmidt orthogonalization process
- Cayley-Hamilton Theorem
- $$\dim(\text{Row Space}) = \dim(\text{Col Space}) = \text{rank}(A)$$
- Spectral Theorem for symmetric matrices
- Invertible Matrix Theorem equivalences
- $$A = PDP^{-1}$$
1Medium
Let \(A\) be a \(4 \times 4\) matrix with eigenvalues \(1, 2, 3, 4\). Compute \(\det(A^3 - 5A^2 + 6A)\).
Show Hint
Factor the polynomial \(p(A) = A(A^2 - 5A + 6I) = A(A - 2I)(A - 3I)\). Then use the fact that \(\det(p(A)) = \prod p(\lambda_i)\).
2Easy
Find the rank and nullity of the matrix \(A = \begin{pmatrix} 1 & 2 & 3 & 4 \\ 2 & 4 & 6 & 8 \\ 1 & 3 & 5 & 7 \end{pmatrix}\). Describe the null space explicitly.
Show Hint
Row-reduce \(A\) to echelon form. Notice that Row 2 is a scalar multiple of Row 1. Identify the free variables and express the null space in parametric vector form.
3Hard
Prove that if \(A\) is an \(n \times n\) real matrix such that \(A^3 = A\), then \(A\) is diagonalizable.
Show Hint
The minimal polynomial of \(A\) divides \(x^3 - x = x(x - 1)(x + 1)\), which has all distinct roots. A matrix is diagonalizable if and only if its minimal polynomial has no repeated roots.
4Medium
Let \(T: P_2 \to P_2\) be defined by \(T(p(x)) = p(x+1) - p(x)\). Find the matrix of \(T\) with respect to the basis \(\{1, x, x^2\}\), its eigenvalues, and determine if \(T\) is diagonalizable.
Show Hint
Compute \(T(1)\), \(T(x)\), and \(T(x^2)\) and express each as a linear combination of the basis. Note that \(T\) lowers the degree, so the matrix is upper triangular.
5Hard
Let \(A\) be a real symmetric \(n \times n\) matrix with eigenvalues \(\lambda_1 \geq \lambda_2 \geq \ldots \geq \lambda_n\). Prove that for any unit vector \(\mathbf{x}\), \(\lambda_n \leq \mathbf{x}^T A \mathbf{x} \leq \lambda_1\), and characterize when equality holds.
Show Hint
Use the spectral decomposition \(A = Q\Lambda Q^T\) and substitute \(\mathbf{y} = Q^T \mathbf{x}\). Since \(Q\) is orthogonal, \(\mathbf{y}\) is also a unit vector. Then \(\mathbf{x}^T A \mathbf{x} = \sum \lambda_i y_i^2\) is a convex combination of eigenvalues.
6Medium
Compute the eigenvalues and eigenvectors of \(A = \begin{pmatrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 6 & -11 & 6 \end{pmatrix}\). Is \(A\) diagonalizable? If so, find \(P\) and \(D\) such that \(A = PDP^{-1}\).
Show Hint
This is a companion matrix. The characteristic polynomial can be read directly from the last row: \(\lambda^3 - 6\lambda^2 + 11\lambda - 6 = (\lambda - 1)(\lambda - 2)(\lambda - 3)\).
7Easy
Apply the Gram-Schmidt process to the vectors \(\mathbf{v}_1 = (1, 1, 0)\), \(\mathbf{v}_2 = (1, 0, 1)\), \(\mathbf{v}_3 = (0, 1, 1)\) to obtain an orthonormal basis for \(\mathbb{R}^3\).
Show Hint
Start with \(\mathbf{u}_1 = \mathbf{v}_1 / \|\mathbf{v}_1\|\). Then subtract projections: \(\mathbf{u}_2' = \mathbf{v}_2 - (\mathbf{v}_2 \cdot \mathbf{u}_1)\mathbf{u}_1\), normalize, and repeat for \(\mathbf{v}_3\).
8Hard
Let \(A\) be an \(n \times n\) matrix such that \(A^2 = I\) (an involution). Prove that \(\mathbb{R}^n = E_1 \oplus E_{-1}\), where \(E_\lambda\) is the eigenspace for eigenvalue \(\lambda\). Express the projections onto \(E_1\) and \(E_{-1}\) in terms of \(A\).
Show Hint
Write any vector \(\mathbf{v}\) as \(\mathbf{v} = \frac{1}{2}(\mathbf{v} + A\mathbf{v}) + \frac{1}{2}(\mathbf{v} - A\mathbf{v})\). Show the first summand is in \(E_1\) and the second is in \(E_{-1}\). The projections are \(P_1 = \frac{1}{2}(I + A)\) and \(P_{-1} = \frac{1}{2}(I - A)\).
9Medium
Let \(V = \{p(x) \in P_3 : p(1) = 0 \text{ and } p'(0) = 0\}\). Show that \(V\) is a subspace of \(P_3\), find its dimension, and give a basis for \(V\).
Show Hint
Write \(p(x) = a + bx + cx^2 + dx^3\). The conditions give \(a + b + c + d = 0\) and \(b = 0\). So \(b = 0\) and \(a = -c - d\). The subspace is 2-dimensional.
10Hard
Let \(A\) be an \(n \times n\) matrix with characteristic polynomial \((\lambda - 2)^3(\lambda - 5)^2\). If \(\text{rank}(A - 2I) = n - 2\) and \(\text{rank}(A - 5I) = n - 1\), find the Jordan normal form of \(A\).
Show Hint
The geometric multiplicity of \(\lambda = 2\) is \(n - \text{rank}(A - 2I) = 2\), so there are 2 Jordan blocks for \(\lambda = 2\). Since algebraic multiplicity is 3, the blocks are sizes \(2 \times 2\) and \(1 \times 1\). For \(\lambda = 5\), geometric multiplicity is 1, so one \(2 \times 2\) block.
11Medium
Let \(A\) and \(B\) be \(n \times n\) matrices. Prove that \(AB\) and \(BA\) have the same eigenvalues (including multiplicity of nonzero eigenvalues). Give a counterexample showing the zero eigenvalue can have different geometric multiplicities.
Show Hint
For \(\lambda \neq 0\): if \(AB\mathbf{v} = \lambda \mathbf{v}\), then \(BA(B\mathbf{v}) = B(AB\mathbf{v}) = \lambda(B\mathbf{v})\), so \(\lambda\) is an eigenvalue of \(BA\) too. For the counterexample, try \(A = \begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix}\) and \(B = \begin{pmatrix} 0 & 0 \\ 0 & 0 \end{pmatrix}\)... or more simply, a \(1 \times 2\) and \(2 \times 1\) matrix.
12Hard
Let \(A\) be an \(n \times n\) real matrix such that \(A^T A = A A^T\) (i.e., \(A\) is normal). Prove that \(A\) and \(A^T\) have the same null space, and conclude that the row space equals the column space of \(A\).
Show Hint
Show \(\|A\mathbf{x}\| = \|A^T \mathbf{x}\|\) for all \(\mathbf{x}\) by expanding \(\mathbf{x}^T A^T A \mathbf{x} = \mathbf{x}^T A A^T \mathbf{x}\). Then \(A\mathbf{x} = \mathbf{0}\) iff \(A^T \mathbf{x} = \mathbf{0}\). The result about row space and column space follows from orthogonal complement relations.