Determinants Practice Test

Advanced Algebra Practice Test: ACT math skills.

Determinants Practice Test

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Determinant Questions · Linear Algebra

Advanced Algebra Practice Test

A determinant is one number that reveals the structure of a square matrix.

This free Determinants Practice Test contains 20 multiple-choice questions and does not require registration. The questions focus on evaluating determinants, tracking signs, using minors and cofactors, applying determinant properties, recognizing singular and invertible matrices, connecting determinants to geometric scale, and using determinant reasoning in systems of equations. Each question has four answer choices, one correct answer, and a worked explanation that identifies the governing rule before carrying out the calculation.

Direct Rules

Cofactors

Properties

Invertibility

Systems

Atlas 01

Start with what a determinant is allowed to describe

A determinant is defined for a square matrix and returns a scalar. It is not another matrix. Its value summarizes whether the associated linear transformation preserves dimension, how it scales area or volume, and whether it reverses orientation.

Magnitude and sign carry different information

The absolute value gives the area or volume scale factor. A negative sign indicates an orientation reversal, while a zero result means the transformation collapses the region into a lower dimension.

Atlas 02

The two-by-two rule is a difference of diagonal products

For a two-row square matrix, multiply along the main diagonal, multiply along the other diagonal, and subtract the second product from the first.

General rule

\det ([\begin{matrix} a & b \\ c & d \end{matrix}]) = a d - b c

The order matters: main-diagonal product minus other-diagonal product.

Worked example

A = [\begin{matrix} 3 & - 2 \\ 5 & 4 \end{matrix}]

\det (A) = 3 \cdot 4 - (- 2) \cdot 5 = 22

The negative entry makes the second diagonal product negative, so subtracting it increases the result.

Atlas 03

Sign discipline is more important than speed

Many determinant errors are not multiplication errors. They come from reversing the two diagonal products, losing a negative entry, or applying the cofactor sign pattern incorrectly.

Write the subtraction first

Keep the two diagonal products visibly separated until both have been evaluated.

Use parentheses around negatives

A negative entry can change the sign of an entire product before the final subtraction occurs.

Estimate the sign

Before simplifying, decide whether each diagonal product is positive or negative. This catches many slips.

Atlas 04

Minors reduce the problem; cofactors attach the correct sign

To form the minor associated with an entry, delete that entry's row and column and take the determinant of what remains. The cofactor is the minor multiplied by its alternating sign.

Cofactor sign field

The upper-left position begins with a positive sign, and the pattern alternates across every row and column.

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Definitions

C_{i j} = {(- 1)}^{i + j} M_{i j}

Here the minor $M_{i j}$ is unsigned. The cofactor $C_{i j}$ includes the sign determined by the row and column position.

Atlas 05

Cofactor expansion converts a larger determinant into smaller ones

You may expand along any row or column. The best choice is usually the one containing the most zeros because every zero eliminates a full minor calculation.

Expansion model

\det (B) = b_{11} C_{11} + b_{12} C_{12} + b_{13} C_{13}

When the cofactor already contains its alternating sign, the expansion itself is written as a sum of entry-cofactor products.

Atlas 06

A carefully chosen expansion can make a three-by-three determinant short

The first row in this example contains a zero, so expanding across that row requires only two smaller determinants.

Matrix

B = [\begin{matrix} 2 & 1 & 0 \\ - 1 & 3 & 2 \\ 4 & 0 & 1 \end{matrix}]

Across the first row, the signs are positive, negative, positive.

Expansion and result

\det (B) = 2 (3 \cdot 1 - 2 \cdot 0) - 1 ((- 1) \cdot 1 - 2 \cdot 4)

\det (B) = 6 - (- 9) = 15

The zero entry contributes nothing, but its position still belongs to the sign pattern.

Atlas 07

Triangular structure replaces expansion with one diagonal product

For an upper-triangular, lower-triangular, or diagonal matrix, the determinant equals the product of the main-diagonal entries. Entries on the other side of the diagonal do not change this rule.

Structural shortcut

T = [\begin{matrix} 4 & - 2 & 7 \\ 0 & 3 & 5 \\ 0 & 0 & - 2 \end{matrix}]

Diagonal product

\det (T) = 4 \cdot 3 \cdot (- 2) = - 24

Recognizing triangular form is both faster and safer than expanding by cofactors.

Atlas 08

Every elementary row operation has a predictable determinant effect

Row reduction can simplify a determinant, but the recorded changes must be reversed or incorporated. Treat the operation history as a ledger.

01Swap two rowsThe determinant changes sign.

02Scale one rowMultiplying one row by

k

multiplies the determinant by

k

03Replace one rowAdding a multiple of another row to a row leaves the determinant unchanged.

04Reach triangular formMultiply the diagonal entries, then account for every swap and row scaling used.

Atlas 09

A zero determinant signals dependence or collapse

Several visible matrix patterns force the determinant to be zero. Recognizing them can eliminate unnecessary calculation.

A zero row or column

Every term in a suitable expansion contains a zero factor.

Repeated rows or columns

Swapping the identical pair would both preserve the matrix and reverse the determinant's sign, forcing a zero result.

Proportional dependence

If one row or column is a scalar multiple of another, the represented directions are not independent.

\det (A) = 0 \Rightarrow A is singular

Atlas 10

The invertibility decision is a zero-versus-nonzero fork

For a square matrix, a nonzero determinant means no dimension is lost. A zero determinant means the transformation collapses at least one direction and cannot be reversed uniquely.

One calculation answers several structural questions

A nonzero determinant is equivalent to full rank, an inverse, a pivot in every row and column, and a unique solution to every compatible square system with that coefficient matrix.

Zero branch

\det (A) = 0

The matrix is singular. Depending on the constants, the related system can have no solution or infinitely many solutions.

Nonzero branch

\det (A) \neq 0

The matrix is invertible. The square system $A x = b$ has exactly one solution for every constant vector $b$ .

Atlas 11

Determinant identities turn long calculations into structural reasoning

When matrices are transposed, multiplied, inverted, or uniformly scaled, use the relevant identity before expanding entries.

Transpose

\det (A^{T}) = \det (A)

. Transposition preserves the determinant.

Product

\det (A B) = \det (A) \det (B)

. This remains true even though matrix multiplication is not commutative.

Inverse

\det (A^{- 1}) = \frac{1}{\det (A)}

, provided the determinant is nonzero.

Scalar

For a square matrix of order

n

\det (k A) = k^{n} \det (A)

. The scalar multiplies every row.

Atlas 12

Scaling the whole matrix is not the same as scaling one row

A common distractor multiplies the determinant by the scalar only once. Scaling the entire matrix scales every row, so the exponent depends on the order of the square matrix.

One row is scaled

\det (A) \to k \det (A)

Only one row contributes the new factor.

The whole matrix is scaled

\det (k A) = k^{n} \det (A)

All $n$ rows contribute one factor of $k$ .

Atlas 13

Parameter questions convert invertibility into an equation

If a matrix contains a parameter, first write its determinant as an algebraic expression. Set that expression equal to zero only when the question asks when the matrix is singular or noninvertible.

Singular-value example

P = [\begin{matrix} p & 4 \\ 3 & p \end{matrix}]

\det (P) = p^{2} - 12

p^{2} - 12 = 0 \Rightarrow p = \pm 2 \sqrt{3}

The matrix is singular at those two parameter values and invertible for every other real value of $p$ .

Atlas 14

Cramer's Rule uses replacement-column determinants to solve a square system

The rule applies when the coefficient determinant is nonzero. Each numerator determinant is formed by replacing one coefficient column with the constants while preserving the other columns and their order.

Column order must stay fixed

The constants replace exactly one variable column at a time. Reordering columns changes the determinant's sign and changes which unknown the quotient represents.

Atlas 15

A full Cramer's Rule check includes the denominator and both replacements

Consider a two-equation system. The coefficient determinant confirms that a unique solution exists before either quotient is formed.

System and coefficient determinant

\begin{matrix} 2 x + y = 7 \\ x - 3 y = - 8 \end{matrix}

D = \det ([\begin{matrix} 2 & 1 \\ 1 & - 3 \end{matrix}]) = - 7

Because $D \neq 0$ , the system has one solution.

Replacement determinants

D_{x} = \det ([\begin{matrix} 7 & 1 \\ - 8 & - 3 \end{matrix}]) = - 13

D_{y} = \det ([\begin{matrix} 2 & 7 \\ 1 & - 8 \end{matrix}]) = - 23

x = \frac{13}{7} y = \frac{23}{7}

Atlas 16

Skills Covered

This medium-level Advanced Algebra review develops both reliable computation and structural judgment. The goal is to recognize when a short property is better than a long expansion.

Direct evaluation

Calculate two-by-two and three-by-three determinants with controlled signs and accurate arithmetic.

Minors and cofactors

Delete the correct row and column, apply the alternating sign, and expand efficiently.

Structural shortcuts

Use triangular form, dependent rows, and elementary row-operation effects.

Determinant identities

Reason about products, transposes, inverses, and scalar multiples without entry-by-entry expansion.

Invertibility

Connect a zero or nonzero determinant to singularity, rank, inverses, and solution behavior.

Systems

Construct replacement determinants and apply Cramer's Rule only when its denominator is nonzero.

Atlas 17

How to Approach the Test

Classify the determinant before calculating. A direct rule, a sparse expansion, a triangular shortcut, or an identity may reduce the work substantially.

01Confirm the matrix is squareA determinant is not defined for a rectangular matrix.

02Inspect the structureLook for zeros, triangular form, repeated or proportional rows, products, transposes, or scalar multiples.

03Select the shortest valid ruleUse the two-by-two formula, a strategic cofactor expansion, row reduction with a ledger, or a determinant identity.

04Track every signSeparate entry signs, cofactor signs, row-swap signs, and the subtraction in the direct formula.

05Answer the actual conditionDistinguish a requested value from a singularity condition, an invertibility decision, or a system solution.

Atlas 18

Common Mistakes

The most common wrong answers can usually be traced to one missed structural check or one unrecorded sign change.

Using a determinant on a rectangular matrix

Check the dimensions first. Only square matrices have determinants.

Reversing diagonal-product order

For a two-by-two determinant, subtract the other-diagonal product from the main-diagonal product.

Confusing a minor with a cofactor

The minor is the smaller determinant; the cofactor also includes the alternating positional sign.

Expanding along mixed rows and columns

Choose one complete row or one complete column and keep the corresponding positions aligned.

Ignoring a row swap

Every single swap reverses the determinant's sign. Two swaps restore the original sign.

Scaling the whole matrix only once

For an order- $n$ matrix, the full scaling factor is $k^{n}$ , not merely $k$ .

Calling every zero determinant inconsistent

A singular coefficient matrix can lead to no solution or infinitely many solutions, depending on the constants.

Using Cramer's Rule with a zero denominator

The coefficient determinant must be nonzero before the determinant quotients are valid.

Stopping at an intermediate determinant

If the question asks for an unknown, an inverse condition, or a parameter value, complete that final step.

Final determinant audit

Run this compact check before accepting a calculation or selecting an answer choice.

Structure → Rule → Sign → Meaning

Is the matrix square?If not, the determinant is undefined.

Was the most efficient method selected?Check for triangular form, zeros, dependence, or a useful identity.

Were all negative entries parenthesized?Keep entry signs separate from cofactor and subtraction signs.

Were row-operation effects recorded?Swaps, row scaling, and row replacement affect the determinant differently.

Does zero or nonzero have the correct interpretation?Connect the result to singularity, invertibility, and system behavior.

Was the requested quantity actually found?Finish parameter solving or Cramer's Rule quotients when required.

Use this free 20-question practice test for Advanced Algebra or introductory Linear Algebra review, placement preparation, or classroom practice. You can retake the test without creating an account. The examples in this review block are illustrative and are not copies of the test questions.