Determinants Practice Quiz with a Step-by-Step Interactive Lesson
Use the quiz at the top of the page 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.
How this determinants practice works
1. Take the quiz: answer the determinant questions at the top of the page.
2. Open the lesson (optional): review how to compute determinants using formulas, cofactors, and row operations.
3. Retry: return to the quiz 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)
Scale a row by \(k\) \(\Rightarrow\) determinant scales by \(k\)
Add a multiple of one row to another \(\Rightarrow\) determinant unchanged
Special matrices, products, and invertibility
Diagonal/triangular matrices: determinant is the product of diagonal entries
Product rule: \(\det(AB)=\det(A)\det(B)\)
Invertibility test: \det(A)≠ 0 and \(\det(A^{-1})=\dfrac{1}{\det(A)}\)
Back to the quiz
When you are ready, return to the quiz at the top of the page and keep practicing determinants.
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Matrix Determinants
Step-by-Step Guide
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Determinants Lesson
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Lesson Overview
Lesson overview
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.
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