The Timpson Lyceum · Linear Algebra (2026)

3.1 Exercises

Problem sheet · 20 points · 4 extra credit

Introduction to Determinants

1. 1. (2 pts) Compute the determinant of $\begin{bmatrix} 5 & 3 \\ -2 & 4 \end{bmatrix}$.

2. 2. (2 pts) Compute $\det A$ by a cofactor expansion across the first row, where

   $A = \begin{bmatrix} 1 & 0 & 2 \\ 3 & -1 & 4 \\ 2 & 1 & 5 \end{bmatrix}.$

3. 3. (3 pts) Compute the determinant of

   $B = \begin{bmatrix} 2 & 0 & 0 & 4 \\ 1 & 3 & 0 & 0 \\ 0 & 5 & -1 & 0 \\ 6 & 0 & 0 & 2 \end{bmatrix}$

   by choosing a row or column that keeps the arithmetic short. State which one you expanded along and why.

4. 4. (2 pts) Without doing any expansion, write down $\det T$ for the triangular matrix

   $T = \begin{bmatrix} 4 & 9 & -3 \\ 0 & 2 & 7 \\ 0 & 0 & 5 \end{bmatrix}.$

5. 5. (3 pts) For the matrix $A$ of Problem 2, recompute $\det A$ by expanding along the second column. Confirm you get the same value, and explain in one sentence why any row or column must give the same answer.

6. 6. (3 pts) State whether each of the following is true or false, and justify each answer.

   (a) The $(i,j)$-cofactor of a matrix equals the determinant of its $(i,j)$ submatrix $A_{ij}$.

   (b) If a square matrix has a row of zeros, its determinant is $0$.

   (c) $\det A$ is defined only for square matrices.

7. 7. (3 pts) Let $A$ be a $3\times 3$ matrix with a row of all zeros. Show, using a cofactor expansion, that $\det A = 0$.

8. 8. (2 pts) Find all values of $t$ for which $\det\begin{bmatrix} 2-t & 1 \\ 1 & 2-t \end{bmatrix} = 0$.

9. 9. (4 pts ••) Prove that the determinant of an $n\times n$ upper-triangular matrix equals the product of its diagonal entries. Use induction on $n$ together with a cofactor expansion along the first column.