Linear Algebra

Linear algebra questions become easier when dimensions and pivots are tracked explicitly. A matrix equation is a system of linear constraints; row operations preserve its solution set when performed correctly. Eigenvalue calculations rely on the characteristic polynomial, while LU decomposition records elimination in a reusable factorization.

Matrices and determinants

Idea. An $m\times n$ matrix has m rows and n columns. Matrix products $AB$ exist when the column count of A equals the row count of B, and generally $AB\ne BA$. For a square matrix, determinant is zero exactly when the matrix is singular; for a $2\times2$ matrix $\begin{bmatrix}a&b\\c&d\end{bmatrix}$, determinant is $ad-bc$.

Solved examples.

1. Question. Find the product $AB$ for $A=\begin{bmatrix}1&2\\0&3\end{bmatrix}$ and $B=\begin{bmatrix}2&1\\4&-1\end{bmatrix}$. Solution. (1) Entry (1,1): $1(2)+2(4)=10$. (2) Entry (1,2): $1(1)+2(-1)=-1$. (3) Second row gives 12 and -3. Answer. $AB=\begin{bmatrix}10&-1\\12&-3\end{bmatrix}$. Trap. Row times column determines each entry.
2. Question. Compute determinant of $\begin{bmatrix}5&2\\3&1\end{bmatrix}$. Solution. (1) Use $ad-bc$. (2) $5(1)-2(3)=5-6=-1$. Answer. -1. Shortcut. A nonzero determinant means the square matrix is invertible.
3. Question. Find determinant of $\begin{bmatrix}1&2&0\\0&3&4\\0&0&-2\end{bmatrix}$. Solution. (1) Matrix is upper triangular. (2) Determinant is product of diagonal entries. (3) $1\cdot3\cdot(-2)=-6$. Answer. -6. Trap. The diagonal-product shortcut requires triangular form.
4. Question. If a $3\times3$ matrix has two equal rows, what is its determinant? Solution. (1) Swapping those rows leaves the matrix unchanged. (2) A row swap changes determinant's sign. (3) Thus det equals its negative and must be zero. Answer. 0. Useful argument. Equal rows imply linear dependence.
5. Question. A matrix has eigenvalues 2, -1, 4. Find its determinant, assuming it is a square matrix over a field and these are all eigenvalues with algebraic multiplicity. Solution. (1) Determinant equals product of eigenvalues. (2) $2(-1)(4)=-8$. Answer. -8. Trap. Include multiplicity; a listed subset of eigenvalues is insufficient.

Systems of linear equations

Idea. A system $Ax=b$ is consistent when b lies in the column space of A. Gaussian elimination uses elementary row operations to expose pivots and free variables. For a square matrix, a unique solution exists exactly when A is invertible, equivalently full rank and nonzero determinant.

Solved examples.

1. Question. Solve $x+y=5$, $x-y=1$. Solution. (1) Add equations: $2x=6$, so x=3. (2) Substitute into first: y=2. Answer. $(x,y)=(3,2)$. Shortcut. Addition eliminates y.
2. Question. Determine consistency of $x+y=2$ and $2x+2y=5$. Solution. (1) Doubling the first equation gives $2x+2y=4$. (2) The second requires the same left side to equal 5. (3) Contradiction. Answer. Inconsistent; no solution. Trap. A zero row with nonzero right-hand side signals inconsistency.
3. Question. Solve $x+2y=5$, $2x+4y=10$ and describe the solutions. Solution. (1) Second equation is exactly twice the first. (2) There is one independent constraint for two variables. (3) Let y=t; then x=5-2t. Answer. Infinitely many solutions $(5-2t,t)$ for real t. Trap. Dependent equations do not determine a unique point.
4. Question. Find k for which the system $x+y=1$, $2x+2y=k$ is consistent. Solution. (1) Twice the first equation gives $2x+2y=2$. (2) It matches the second exactly only if k=2. Answer. k=2, with infinitely many solutions; otherwise none. Trap. State both the parameter condition and the solution count.
5. Question. Solve $x+y+z=6$, $y+z=4$, $z=1$. Solution. (1) Third equation gives z=1. (2) Then y=3. (3) First gives x=2. Answer. (2,3,1). Shortcut. Triangular systems are solved by back substitution.

Eigenvalues and eigenvectors

Idea. A nonzero vector v is an eigenvector of square A when $Av=\lambda v$; λ is its eigenvalue. Eigenvalues solve $\det(A-\lambda I)=0$. Eigenvectors for λ are nonzero solutions to $(A-\lambda I)v=0$. Repeated eigenvalues need not have equally many independent eigenvectors.