Determinants by Expansion Practice Test

Advanced Algebra Practice Test: ACT math skills.

Determinants by Expansion Practice Test

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Determinant Questions · Linear Algebra

Expansion turns one determinant into a controlled tree of smaller determinants.

This free Determinants by Expansion Practice Test contains 20 multiple-choice questions and does not require registration. The questions focus on minors, cofactors, alternating signs, row and column expansion, zero-rich lines, recursive reduction, symbolic entries, higher-order determinants, row-operation preparation, invertibility, and system conditions. Each question has four answer choices, one correct answer, and a worked explanation that shows which expansion route to choose, which branches survive, and how the smaller determinants combine.

Choose

Delete

Sign

Expand

Reduce

Interpret

Branch 01

Expansion is a method of reduction, not a new kind of determinant

Laplace expansion rewrites a determinant of order $n$ as a weighted sum of determinants of order $n - 1$ . Repeating the process eventually reaches two-by-two determinants.

Select one line

Choose a complete row or a complete column.

Create one minor per entry

Delete the entry's row and column to expose the smaller determinant.

Assemble signed branches

Multiply entries by cofactors and add all contributions from the selected line.

Branch 02

Each selected entry opens exactly one cofactor branch

A three-entry row produces three possible contributions, but a zero entry makes its entire branch equal to zero.

Zeros save more than one multiplication

A zero entry removes the need to build, evaluate, sign, and multiply its minor. That is why line choice matters.

Branch 03

The minor is determined by the entry's exact position

For position $(i, j)$ , remove row $i$ and column $j$ . The remaining square array defines $M_{i j}$ .

Source matrix

A = [\begin{matrix} 2 & 0 & 1 \\ 3 & - 1 & 4 \\ 0 & 5 & 2 \end{matrix}]

Focus on the zero in the first row and second column.

Its minor

M_{12} = \det ([\begin{matrix} 3 & 4 \\ 0 & 2 \end{matrix}]) = 6

The minor exists and equals six even though the selected matrix entry is zero.

Branch 04

The cofactor adds the alternating sign to the minor

Do not attach the sign twice. Either use signed cofactors in a sum or write the alternating signs directly in the expansion.

Sign field

The upper-left position starts positive and every horizontal or vertical step flips the sign.

KEEPFLIPKEEPFLIPFLIPKEEPFLIPKEEPKEEPFLIPKEEPFLIPFLIPKEEPFLIPKEEP

Cofactor rule

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

C_{12} = - M_{12} = - 6

The branch contribution is still zero because $a_{12} C_{12} = 0 (- 6) = 0$ .

Branch 05

Row expansion and column expansion are parallel formulas

Fix a row index and sum across columns, or fix a column index and sum down rows. Never mix positions from several lines.

Expand across row $i$

\det (A) = \sum_{j = 1}^{n} a_{i j} C_{i j}

Expand down column $j$

\det (A) = \sum_{i = 1}^{n} a_{i j} C_{i j}

Branch 06

First-row expansion leaves two active branches

The middle entry is zero, so only the first and third positions need minor calculations.

Branch expression

\det (A) = 2 ((- 1) (2) - 4 (5)) + 1 (3 (5) - (- 1) (0))

Simplify after assembly

\det (A) = 2 (- 22) + 15 = - 29

The positive sign at the third position is already reflected in the plus sign between the active contributions.

Branch 07

A different expansion line must produce the same determinant

Expanding the same matrix down its first column creates a different-looking calculation but the same scalar result.

\det (A) = 2 (- 22) - 3 (- 5) + 0 (M_{31}) = - 29

Route independence

Every complete row or column is valid.

Calculation differences

The minors and intermediate signs depend on the selected line.

Audit use

A second sparse line can provide a strong verification when time permits.

Branch 08

Choose the line that minimizes active work

Count nonzero entries first. If two lines tie, prefer the one with simpler numbers or minors that are triangular.

01Count zerosEach zero removes a complete cofactor branch.

02Compare entry sizesSmall entries usually make arithmetic easier and reduce large intermediate values.

03Preview the minorsPrefer positions that expose zeros, repeated patterns, or triangular submatrices.

04Check the first signIf the selected line does not begin at a positive position, write the correct alternating sequence before calculating.

Branch 09

Higher-order expansion is recursive

A four-by-four determinant expands into three-by-three determinants, which can expand into two-by-two determinants. Sparse lines keep this recursion under control.

Expand only as far as needed

At each level, rescan the new minor for zeros or triangular form. Do not automatically repeat a long dense expansion.

Branch 10

A four-by-four matrix with three zeros can collapse to one branch

Expanding across the first row leaves only the upper-left contribution.

Sparse source

D = [\begin{matrix} 1 & 0 & 0 & 0 \\ 2 & 3 & 1 & 4 \\ 0 & - 1 & 2 & 5 \\ 6 & 0 & 3 & 1 \end{matrix}]

Surviving minor

\det (D) = \det ([\begin{matrix} 3 & 1 & 4 \\ - 1 & 2 & 5 \\ 0 & 3 & 1 \end{matrix}]) = - 50

The first-row coefficient and cofactor sign are both positive.

Branch 11

Row replacement can create a better expansion line without changing the determinant

Before expanding a dense determinant, consider adding a multiple of one row to another to introduce zeros. Record any swaps or row scaling because those operations do change the determinant.

Row replacement

Adding a multiple of one row to another preserves the determinant.

Row swap

Swapping two rows reverses the determinant's sign.

Row scaling

Multiplying one row by $k$ multiplies the determinant by $k$ .

Branch 12

Symbolic expansion produces an algebraic condition

Choose the third column in this example. Its first two entries are zero, so the determinant becomes one factor times a two-by-two minor.

Symbolic matrix

P = [\begin{matrix} q & 2 & 0 \\ 1 & q + 1 & 0 \\ 0 & 0 & q - 2 \end{matrix}]

Factorized determinant

\det (P) = (q - 2) (q^{2} + q - 2)

\det (P) = (q - 2) (q + 2) (q - 1)

The matrix is singular when $q = 2$ , $q = - 2$ , or $q = 1$ .

Branch 13

Expansion can answer an invertibility question without finding an inverse

For a square matrix, the determinant only needs to be classified as zero or nonzero.

Zero result

\det (A) = 0

The matrix is singular and cannot be inverted.

Nonzero result

\det (A) \neq 0

The matrix is invertible and a related square system has one unique solution.

Branch 14

Skills Covered

These medium-level Advanced Algebra questions test expansion mechanics, strategy, and interpretation.

Minor construction

Remove the correct row and column while preserving the order of surviving entries.

Cofactor signs

Combine each minor with its alternating positional sign.

Line selection

Choose the row or column that eliminates the most work.

Recursive reduction

Reduce higher-order determinants until a direct base calculation is available.

Symbolic expansion

Factor the determinant expression and solve singularity or invertibility conditions.

Structural reasoning

Use row operations, sparse patterns, and a second expansion route as checks.

Branch 15

How to Approach the Test

Treat every determinant as a branching decision before treating it as an arithmetic problem.

01Inspect all rows and columnsCount zeros and look for simple entries or easy minors.

02Mark the sign sequenceStart from the actual first position on the selected line.

03Prune zero branchesDo not calculate a minor when its entry multiplier is zero.

04Build each active minorDelete the correct row and column and evaluate the smaller determinant separately.

05Assemble and interpretAdd the signed contributions, then answer the requested value or condition.

Branch 16

Common Mistakes

Expansion errors usually arise from a wrong branch definition, a lost sign, or an incomplete return from the minor to the original determinant.

Mixing entries from several lines

An expansion must use one complete row or one complete column.

Deleting the wrong row or column

The minor must remove the row and column of the selected entry.

Restarting every sign pattern as positive

The first sign depends on the first selected position, especially for an interior row or column.

Applying the cofactor sign twice

Do not alternate signs again when signed cofactors are already being used.

Calculating a zero branch

A zero entry makes the whole entry-cofactor product zero regardless of its minor.

Assuming a zero entry has a zero minor

The minor depends on the surviving submatrix, not on the selected entry's value.

Forgetting the entry multiplier

Each minor must be multiplied by both its selected entry and positional sign.

Ignoring row-operation effects

Row replacement preserves the determinant, but swaps and row scaling do not.

Stopping at a smaller determinant

Return every evaluated minor to the full expansion before reporting the answer.

Final branch audit

Trace each surviving contribution from the selected line to the final scalar.

Choose → Prune → Reduce → Recombine

Was one complete row or column selected?All contributions must come from the same expansion line.

Were zero branches removed before calculation?A zero entry eliminates its full cofactor contribution.

Does every active minor delete the correct lines?Verify the surviving entry order before evaluating.

Were positional signs applied exactly once?Separate cofactor signs from negative entries inside minors.

Were all smaller results returned to the main sum?No minor by itself is the original determinant.

Was the final determinant interpreted?Complete any invertibility, parameter, or system conclusion.

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.

Determinants by Expansion Practice Test