3x3 Determinants Practice Test

Advanced Algebra Practice Test: ACT math skills.

3x3 Determinants Practice Test

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Determinant Questions · Linear Algebra

Advanced Algebra Practice Test

A larger determinant becomes manageable when you cut it into smaller pieces.

This free Three-by-Three Determinants Practice Test contains 20 multiple-choice questions and does not require registration. The questions focus on minors, cofactors, sign patterns, cofactor expansion, Sarrus' Rule, triangular shortcuts, row-operation effects, zero determinants, invertibility, parameter values, determinant identities, and systems. Each question has four answer choices, one correct answer, and a worked explanation that shows how to select a method, preserve signs, evaluate smaller determinants, and interpret the final scalar.

Minors

Cofactors

Expansion

Sarrus

Row Rules

Invertibility

Manual 01

Nine entries create several valid expansion routes

A three-by-three determinant is still one scalar, but the direct calculation must combine smaller two-by-two determinants or an equivalent structured method.

Three row routes

You may expand across the first, second, or third row.

Three column routes

You may expand down the first, second, or third column.

One final value

Every correct route produces the same determinant.

Manual 02

A minor is exposed by removing one row and one column

For the entry in row $i$ and column $j$ , delete that entire row and column. The four entries left behind form the two-by-two minor determinant.

Delete complete lines, not nearby entries

The selected entry identifies exactly one row and one column. Removing any other pair changes the minor and therefore changes the cofactor.

Manual 03

Minor notation keeps the selected position visible

Consider a matrix and the position in its second row and third column.

Source matrix

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

Delete row two and column three.

Exposed minor

M_{23} = \det ([\begin{matrix} 1 & 2 \\ 2 & 5 \end{matrix}]) = 5 - 4 = 1

The minor is one before the cofactor sign is applied.

Manual 04

A cofactor combines the minor with an alternating positional sign

The upper-left position starts positive. Signs alternate across each row and down each column.

Alternating sign map

Odd position sums reverse the minor; even position sums keep it.

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Cofactor definition

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

C_{23} = {(- 1)}^{5} (1) = - 1

The minor and cofactor are not always equal; the selected position controls the sign.

Manual 05

Cofactor expansion is a sum of entry-cofactor products

Choose one entire row or column. Multiply every entry on that line by its own cofactor, then add the three contributions.

\det (A) = a_{11} C_{11} + a_{12} C_{12} + a_{13} C_{13}

Entry

Use the value from the selected expansion line.

Minor

Delete that entry's row and column and evaluate what remains.

Cofactor sign

Apply the alternating sign before adding the contribution.

Manual 06

A zero on the expansion line removes one complete calculation

For the sample matrix, expanding across the first row requires only two nonzero contributions.

Choose first row

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

The signs across this row are positive, negative, positive.

Expand and simplify

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

\det (A) = - 21 - 2 (- 5) = - 11

The zero entry contributes zero without requiring its minor.

Manual 07

The best expansion line usually contains the most zeros

Choosing a line is a strategic decision. A row or column with two zeros turns a three-term expansion into one two-by-two calculation.

Zero-rich first column

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

One surviving term

\det (B) = 3 (4 \cdot 6 - (- 1) \cdot 5) = 3 (29) = 87

Expanding across a dense row would be correct but unnecessarily long.

Manual 08

Sarrus' Rule organizes six triple products for this order only

Copy the first two columns to the right, add the three downward diagonal products, and subtract the three upward diagonal products. This shortcut does not extend to larger square matrices.

Use a fixed inventory of terms

There are three products in the forward group and three products in the backward group. Missing or duplicating one product changes the result.

(- 1 + 16 + 0) - (0 + 6 + 20) = - 11

Manual 09

Triangular form reduces the determinant to a diagonal product

For upper-triangular, lower-triangular, or diagonal matrices, multiply the three main-diagonal entries.

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

\det (T) = 2 \cdot (- 4) \cdot 5 = - 40

No cofactor expansion is needed after triangular structure is recognized.

Manual 10

Row reduction works only when every determinant effect is recorded

The goal is often to reach triangular form, but swaps and row scaling must remain in the calculation ledger.

01Swap two rowsReverse the determinant's sign.

02Scale one row by $k$ Multiply the determinant by

k

03Add a multiple of one row to anotherLeave the determinant unchanged.

04Reach triangular formMultiply the diagonal entries, then account for the recorded changes.

Manual 11

Dependence patterns force a zero determinant

Recognizing structural zeros is faster and more reliable than expanding a determinant that must vanish.

Zero row or column

A suitable expansion contains only zero contributions.

Repeated rows or columns

Swapping the identical pair would preserve the matrix but reverse the sign, forcing zero.

Linear dependence

If one row is a combination of the others, the matrix loses a dimension.

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

Manual 12

Parameter values are found from a determinant equation

Use structure before expanding. The example is block triangular, so its determinant is a scalar factor times a two-by-two determinant.

Parameter matrix

P = [\begin{matrix} p & 1 & 0 \\ 2 & p & 0 \\ 0 & 0 & 3 \end{matrix}]

Singularity condition

\det (P) = 3 (p^{2} - 2)

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

The matrix is invertible for every other real value of $p$ .

Manual 13

Order matters when the entire matrix is scaled

A scalar applied to every entry contributes once from each of the three rows. Other determinant identities remain available without expansion.

Transpose

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

Product

$\det (A B) = \det (A) \det (B)$ .

Full scaling

$\det (k A) = k^{3} \det (A)$ .

Manual 14

The absolute determinant is a three-dimensional volume scale factor

The three columns can represent three vectors. Their determinant gives signed volume: magnitude measures the parallelepiped volume, sign records orientation, and zero means the vectors lie in a lower-dimensional space.

Zero has a geometric meaning

If the columns are dependent, the solid collapses into a plane or line and its volume becomes zero.

Manual 15

A nonzero coefficient determinant guarantees one system solution

For a square three-variable system, the coefficient determinant is the structural checkpoint before inverse methods or Cramer's Rule.

Singular coefficient matrix

\det (A) = 0

The system cannot have exactly one solution; the constants determine whether it has none or infinitely many.

Invertible coefficient matrix

\det (A) \neq 0

The system has exactly one solution and inverse or determinant-based solution methods are valid.

Manual 16

Method choice should follow the visible structure

A correct but unnecessarily long method creates more opportunities for sign and arithmetic errors.

Cofactor expansion

Best when a row or column contains zeros or simple entries.

Sarrus' Rule

Useful for a dense numerical matrix when all six products can be tracked reliably.

Row reduction

Useful when simple row replacements can create triangular form without awkward fractions.

Manual 17

Skills Covered

These medium-level Advanced Algebra questions combine calculation with strategic method selection and structural interpretation.

Minor extraction

Delete the correct row and column and evaluate the remaining two-by-two determinant.

Cofactor signs

Apply the alternating positional sign without confusing a minor and a cofactor.

Expansion

Choose one complete row or column and assemble all entry-cofactor contributions.

Shortcuts

Use zeros, triangular form, Sarrus' Rule, or row operations when appropriate.

Invertibility

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

Parameters and identities

Solve determinant conditions and reason about transpose, products, and scaling.

Manual 18

How to Approach the Test

Inspect first, choose a method second, and calculate only after the sign structure is visible.

01Scan for structureLook for zeros, triangular form, repeated rows, proportionality, or simple row operations.

02Select the shortest methodChoose cofactor expansion, Sarrus' Rule, a diagonal shortcut, or tracked row reduction.

03Mark signs before arithmeticWrite the cofactor pattern or separate the forward and backward Sarrus groups.

04Evaluate smaller piecesUse parentheses around negative entries and keep minor calculations separate.

05Interpret the resultFinish any invertibility decision, parameter condition, volume statement, or system conclusion.

Manual 19

Common Mistakes

Most errors come from removing the wrong row or column, losing a positional sign, or mixing parts of different methods.

Deleting only the selected entry

A minor requires removal of the entire row and entire column containing that entry.

Calling every minor a cofactor

The cofactor includes the alternating sign; the minor alone does not.

Expanding along mixed lines

Use one complete row or one complete column for a cofactor expansion.

Starting every expansion with a positive sign

The first sign depends on the first selected position, not on the order in which terms are written.

Using Sarrus' Rule beyond this order

The repeated-column shortcut is specific to three-by-three determinants.

Forgetting the backward Sarrus group

Subtract all three backward products from the sum of all three forward products.

Ignoring row-operation effects

Swaps and row scaling change the determinant even when row reduction simplifies the matrix.

Using only one scalar factor

Scaling the entire matrix by $k$ scales the determinant by $k^{3}$ .

Stopping at an intermediate minor

Return to the full expansion and answer the requested determinant or condition.

Final cutaway audit

Check the route from selected line to final interpretation before choosing an answer.

Select → Cut → Sign → Assemble

Was the shortest valid method chosen?Use zeros and triangular structure whenever possible.

Was one complete row or column used?Do not mix expansion positions from different lines.

Did every minor remove the correct row and column?Recheck the four surviving entries before calculating.

Were cofactor signs applied separately?Keep positional signs distinct from negative matrix entries.

Were row-operation changes recorded?Account for swaps and scaling before accepting a triangular result.

Was the final scalar interpreted?Complete the invertibility, parameter, geometry, 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.