Learn Determinants: Interactive Problems and Worked Examples
Use the question set below to practice determinants and the most important determinant properties you need for Linear Algebra: determinant notation \(\det(A)\) and what it measures (signed area/volume scaling), the must-know \(2\times 2\) determinant formula \(\det\!\begin{pmatrix}a&b\\c&d\end{pmatrix}=ad-bc\), \(3\times 3\) determinants using cofactor (Laplace) expansion and choosing a row/column with zeros, fast methods with row reduction / Gaussian elimination while tracking row operations (swapping rows flips the sign, scaling a row scales the determinant, adding a multiple of one row to another keeps the determinant unchanged), quick determinants of diagonal and triangular matrices (product of diagonal entries), key algebra rules like \(\det(AB)=\det(A)\det(B)\), \(\det(A^T)=\det(A)\), and \(\det(kA)=k^n\det(A)\), and the link between determinant and invertibility (a matrix is invertible iff \(\det(A)≠ 0\)), including permutation matrix determinants (\(\pm 1\)) and sign (even/odd permutations). If you want a refresher, click Start lesson to open a step-by-step guide with worked examples and quick checks.
Answer the question set and review your mistakes at the end.
How this determinants practice works
1. Take the practice set: answer the determinant questions below.
2. Open the lesson (optional): review how to compute determinants using formulas, cofactors, and row operations.
3. Retry: return to the question set and apply determinant rules immediately to improve speed and accuracy.
What you will learn in the determinants lesson
\(2\times 2\) determinants and quick interpretation
Compute \(\det\!\begin{pmatrix}a&b\\c&d\end{pmatrix}=ad-bc\) fast and accurately
Understand \(\det(A)=0\) as a singular matrix and non-invertibility
Connect \(|\det(A)|\) to area scaling in 2D
\(3\times 3\) determinants with cofactors
Use cofactor (Laplace) expansion and the sign pattern \((+,-,+)\)
Choose a row/column with zeros to simplify computations
Spot zero determinants quickly (repeated/proportional rows or columns)
Purpose: Build a clear understanding of determinants so you can compute the determinant of a matrix \(\det(A)\) for \(2\times 2\), \(3\times 3\), and larger square matrices. You’ll learn the \(2\times 2\) formula \(ad-bc\), use cofactor (Laplace) expansion for \(3\times 3\) when it’s efficient, and use row operations / row reduction to simplify a matrix to triangular form while tracking how each operation changes the determinant. You’ll also apply key rules like \(\det(AB)=\det(A)\det(B)\) and \(\det(kA)=k^n\det(A)\), connect determinants to invertibility (singular vs. invertible matrices), and recognize fast patterns (diagonal/triangular matrices, permutation matrices).
Success criteria
Compute \(\det\!\begin{pmatrix}a&b\\c&d\end{pmatrix}=ad-bc\) for \(2\times 2\) matrices.
Compute \(3\times 3\) determinants using cofactor expansion and smart choices (rows/columns with zeros).
Use row operations correctly: swap rows \(\Rightarrow\) sign change; scale a row \(\Rightarrow\) scale the determinant; add a multiple of one row to another \(\Rightarrow\) determinant unchanged.
Compute determinants of triangular and diagonal matrices as the product of diagonal entries.
Apply determinant rules: \(\det(AB)=\det(A)\det(B)\), \(\det(A^T)=\det(A)\), and \(\det(kA)=k^n\det(A)\).
Use \(\det(A)≠ 0\) as an invertibility test and understand \(\det(A^{-1})=\dfrac{1}{\det(A)}\) when \(A\) is invertible.
Recognize fast “zero determinant” signals (repeated/proportional rows or columns, or a zero row/column).
Key vocabulary
Determinant: a scalar \(\det(A)\) associated with a square matrix \(A\); it encodes scaling and orientation of the linear transformation.
Singular / invertible: \(A\) is invertible iff \(\det(A)≠ 0\); if \(\det(A)=0\), \(A\) is singular.
Minor: the determinant of the smaller matrix obtained by deleting a row and column.
Cofactor: \(C_{ij}=(-1)^{i+j}M_{ij}\), where \(M_{ij}\) is the minor determinant.
Laplace (cofactor) expansion: a method to compute \(\det(A)\) by expanding along a chosen row or column.
Permutation matrix: a matrix that permutes the standard basis; its determinant is \(\pm 1\) (sign of the permutation).
Quick pre-check
Pre-check 1: What is the determinant of \(\begin{pmatrix}2 & 5 \\ 7 & 1\end{pmatrix}\)?
Hint: For \(\begin{pmatrix}a & b\\c & d\end{pmatrix}\), \(\det = ad-bc\).
Pre-check 2: What happens to the determinant when two rows of a matrix are swapped?
Hint: A single row swap flips the sign of \(\det(A)\).
2×2 Determinants
Determinant basics: the \(2\times 2\) formula and what it means
Learning goal: Compute \(2\times 2\) determinants quickly and recognize what \(\det(A)=0\) means.
Key idea
For a \(2\times 2\) matrix \[ A=\begin{pmatrix}a & b\\ c & d\end{pmatrix}, \] the determinant is \[ \det(A)=ad-bc. \] This number tells you whether the transformation is invertible (it is invertible iff \(\det(A)≠ 0\)), and in 2D it represents the signed area scaling factor. A negative determinant means orientation is reversed (a “flip”).
Use \(ad-bc\) with \(a=0\), \(b=4\), \(c=9\), \(d=0\): \[ \det(A)=0\cdot 0 - 4\cdot 9 = -36. \] The magnitude \(|-36|=36\) is the area scaling factor; the negative sign indicates a flip in orientation.
Try it
Try it 1: What is the determinant of \(\begin{pmatrix} 2 & 4 \\ 1 & 2 \end{pmatrix}\)?
Hint: Compute \(2\cdot 2-4\cdot 1\).
Try it 2: What is the determinant of \(\begin{pmatrix}0 & 4 \\ 9 & 0\end{pmatrix}\)?
Hint: Use \(ad-bc\). Zeros make the arithmetic quick.
\(\det(A)=0\) means \(A\) is singular (not invertible).
3×3 Determinants
\(3\times 3\) determinants: cofactor expansion and smart choices
Learning goal: Compute \(3\times 3\) determinants accurately and efficiently using cofactors.
Key idea
For a \(3\times 3\) matrix, a reliable method is cofactor (Laplace) expansion. The sign pattern is: \[ \begin{pmatrix} + & - & +\\ - & + & -\\ + & - & + \end{pmatrix}. \] Expanding along a row or column with zeros reduces the work a lot, because any term with a \(0\) entry disappears.
Expand along the third column. The first two entries in that column are \(0\), so only the \((3,3)\) entry contributes: \[ \det(A)=1\cdot \det\!\begin{pmatrix}1 & 2\\3 & 4\end{pmatrix}. \] Now compute the \(2\times 2\) determinant: \[ \det\!\begin{pmatrix}1 & 2\\3 & 4\end{pmatrix}=1\cdot 4-2\cdot 3=4-6=-2. \] So \(\det(A)=-2\).
Try it
Try it 1: What is the determinant of \(\begin{pmatrix}2 & 3 & 5\\1 & 0 & 4\\0 & 1 & 2\end{pmatrix}\)?
Hint: Expand along the first row: \(2\det\!\begin{pmatrix}0&4\\1&2\end{pmatrix}-3\det\!\begin{pmatrix}1&4\\0&2\end{pmatrix}+5\det\!\begin{pmatrix}1&0\\0&1\end{pmatrix}\).
Try it 2: What is the determinant of \(\begin{pmatrix} 1 & 2 & 3 \\ 1 & 2 & 3 \\ 4 & 5 & 6 \end{pmatrix}\)?
Hint: If two rows (or columns) are identical, the determinant is \(0\).
Summary
Use cofactor expansion, and choose a row/column with zeros when possible.
Repeated or proportional rows/columns \(\Rightarrow \det(A)=0\).
Row Operations
Row operations and determinant properties
Learning goal: Use row operations to simplify a matrix while tracking how \(\det(A)\) changes.
Key idea
These three row operations have predictable effects on determinants:
Swap two rows: multiplies the determinant by \(-1\).
Multiply a row by \(k\): multiplies the determinant by \(k\).
Add \(k\) times one row to another: does not change the determinant.
This lets you use elimination to create zeros and then compute the determinant of a triangular matrix as a product of diagonal entries.
Worked example
Example: Compute \(\det\!\begin{pmatrix}1 & 2 & 1\\0 & 3 & 4\\2 & 1 & 0\end{pmatrix}\) using a row operation.
Use \(R_3 \leftarrow R_3 - 2R_1\). This operation does not change the determinant: \[ \begin{pmatrix}1 & 2 & 1\\0 & 3 & 4\\2 & 1 & 0\end{pmatrix} \;\to\; \begin{pmatrix}1 & 2 & 1\\0 & 3 & 4\\0 & -3 & -2\end{pmatrix}. \] Now expand along the first column (two zeros below the top): \[ \det(A)=1\cdot \det\!\begin{pmatrix}3 & 4\\-3 & -2\end{pmatrix} =1\cdot(3(-2)-4(-3))=-6+12=6. \]
Try it
Try it 1: If each row of a \(3 \times 3\) matrix is multiplied by \(2\), by what factor does its determinant change?
Hint: Multiplying one row by \(2\) multiplies \(\det\) by \(2\). Three rows \(\Rightarrow 2^3\).
Try it 2: Which row operation multiplies the determinant of a matrix by \(-1\)?
Hint: One swap flips orientation, so it flips the sign of the determinant.
Summary
Track row swaps and row scalings; use row-addition operations freely (they do not change \(\det\)).
Triangular matrix \(\Rightarrow\) determinant is the product of diagonal entries.
Special Matrices
Diagonal, triangular, and permutation matrices
Learning goal: Recognize fast determinant patterns for special matrices.
Triangular matrix: for upper/lower triangular matrices, \(\det\) is the product of diagonal entries.
Permutation matrix: \(\det(P)=+1\) for an even permutation and \(\det(P)=-1\) for an odd permutation.
Worked example
Example: What is the determinant of \(\begin{pmatrix}5 & 0 & 0\\0 & -2 & 0\\0 & 0 & 3\end{pmatrix}\)?
This matrix is diagonal, so multiply the diagonal entries: \[ \det(A)=5\cdot(-2)\cdot 3=-30. \]
Try it
Try it 1: What is the determinant of \(\begin{pmatrix} 2 & 1 & 0 \\ 0 & 3 & 0 \\ 0 & 0 & 4 \end{pmatrix}\)?
Hint: It’s upper triangular, so multiply the diagonal entries.
Try it 2: What is the determinant of the \(4\times 4\) cyclic permutation matrix for \((1\to 2\to 3\to 4\to 1)\)?
Hint: A 4-cycle is an odd permutation (it can be written as 3 row swaps), so the determinant is \(-1\).
Summary
Diagonal/triangular matrices: determinant is the product of the diagonal entries.
Permutation matrices: determinant is \(\pm 1\) depending on parity (even/odd permutation).
Determinant Rules
Determinant algebra: \(\det(AB)\), \(\det(kA)\), and \(\det(A^{-1})\)
Learning goal: Use the main determinant rules to simplify problems without computing from scratch.
Key idea
Product rule: \(\det(AB)=\det(A)\det(B)\).
Scalar multiple: if \(A\) is \(n\times n\), then \(\det(kA)=k^n\det(A)\).
Inverse: if \(A\) is invertible, then \(\det(A^{-1})=\dfrac{1}{\det(A)}\).
Transpose: \(\det(A^T)=\det(A)\).
Worked example
Example: If \(\det(A)=-2\) for a \(2\times 2\) matrix \(A\), what is \(\det(2A)\)?
Here \(n=2\), so \[ \det(2A)=2^2\det(A)=4(-2)=-8. \]
Try it
Try it 1: If matrices \(A\) and \(B\) are both \(3 \times 3\), what is the determinant of \(AB\)?
Hint: Determinants turn matrix multiplication into multiplication of numbers.
Try it 2: If you swap both the rows and columns of a \(2 \times 2\) matrix, what happens to its determinant?
Hint: Swapping rows flips the sign, and swapping columns flips the sign again. Two sign flips cancel.
Summary
\(\det(AB)=\det(A)\det(B)\).
\(\det(kA)=k^n\det(A)\) for \(n\times n\) matrices.
Zero Determinants
When is \(\det(A)=0\)? Fast checks for singular matrices
Learning goal: Spot determinant \(0\) quickly using structure and row/column relationships.
Key idea
If two rows (or columns) are equal, then \(\det(A)=0\).
If one row (or column) is a multiple of another, then \(\det(A)=0\).
If a row or column is all zeros, then \(\det(A)=0\).
\(\det(A)=0\) means the rows/columns are linearly dependent, so \(A\) is not invertible.
Worked example
Example: What is \(\det\!\begin{pmatrix}2 & 4\\3 & 6\end{pmatrix}\)?
The second row \((3,6)\) is \(\tfrac{3}{2}\) times the first row \((2,4)\), so the rows are dependent. That guarantees \(\det(A)=0\). You can also verify with \(ad-bc\): \[ \det(A)=2\cdot 6 - 4\cdot 3 = 12-12=0. \]
Try it
Try it 1: What is the determinant of \(\begin{pmatrix}2 & 4 \\ 3 & 6\end{pmatrix}\)?
Hint: The rows are proportional, so the determinant must be \(0\).
Try it 2: What happens to the determinant of a \(2 \times 2\) matrix if both rows are multiplied by \(2\)?
Hint: Each row scaling by \(2\) multiplies the determinant by \(2\). Two rows \(\Rightarrow 2^2=4\).
Summary
Dependent rows/columns \(\Rightarrow \det(A)=0\).
Scaling each row by \(k\) scales the determinant by \(k\) each time.
Applications & Big Picture
Why determinants matter
Learning goal: Connect determinants to geometry and to key linear algebra ideas, and finish with a final check.
Where determinants show up
Invertibility: \(A\) is invertible iff \(\det(A)≠ 0\).
Area and volume: \(|\det(A)|\) is an area/volume scaling factor.
Solving systems: determinants appear in Cramer’s rule and formulas involving inverses.
Eigenvalues: the determinant equals the product of eigenvalues (counting multiplicity) for square matrices.
Worked example: area of a parallelogram
Example: Find the area of the parallelogram spanned by vectors \(\vec{u}=(2,1)\) and \(\vec{v}=(3,4)\).
Place the vectors as columns of a matrix: \[ A=\begin{pmatrix}2 & 3\\1 & 4\end{pmatrix}. \] The area is \(|\det(A)|\): \[ |\det(A)|=\left|2\cdot 4 - 3\cdot 1\right|=\left|8-3\right|=5. \] So the area is \(5\).
Try it
Try it 1: Compute the determinant of \(\begin{pmatrix}1 & 2 & 0\\3 & 4 & 0\\5 & 6 & 1\end{pmatrix}\).
Hint: Expand along the third column; only one term survives.
Try it 2: What is true about the determinant of an invertible square matrix?
Next step: Close this lesson and try your quiz again. If you miss a question, reopen the book and review the page that matches the determinant skill you need.
