Algebra Practice

Cofactors Practice Test

Advanced Algebra Practice Test: ACT math skills.

Cofactors Practice Test

This test has 20 questions

Instant feedback · Worked explanations
Determinant Questions · High School Linear Algebra

A cofactor is a minor with the correct position sign attached.

This free Cofactors Practice Test contains 20 multiple-choice questions and does not require registration. The questions are written for high school Advanced Algebra practice and focus on locating matrix entries, deleting the correct row and column, calculating two-by-two minors, applying the alternating sign pattern, finding individual cofactors, building a cofactor matrix, and using cofactors in determinant expansion. Each question has four answer choices, one correct answer, and a detailed explanation that keeps the minor calculation separate from the sign decision.

Locate
Delete
Calculate
Apply Sign
Mosaic 01

A minor and a cofactor are related, but they are not identical

The minor comes from a smaller determinant. The cofactor is that minor multiplied by a sign determined by its original row and column position.

Minor

Mij=determinant after deleting row i and column j

This value does not yet include the checkerboard sign.

Cofactor

Cij=(−1)i+jMij

This is the signed value used in cofactor expansion.

Mosaic 02

The chosen position tells you exactly what to remove

To find the cofactor in row two and column three, delete all of row two and all of column three. Keep the four surviving entries in their original order.

remove selected row and columnminor remains

Two coordinates control the cut

The row coordinate identifies one horizontal line; the column coordinate identifies one vertical line. Their intersection is the original selected entry.

Mosaic 03

The sign pattern alternates like a checkerboard

Start with a positive position in the upper-left corner. Move one step horizontally or vertically and the sign changes.

Sign pattern

Memorize the arrangement, then verify it with row-plus-column parity.

KEEPFLIPKEEPFLIPKEEPFLIPKEEPFLIPKEEP
Parity check
i+j even ⇒ keep the minor's sign
i+j odd ⇒ reverse the minor's sign

This rule works for any cofactor position and prevents guessing.

Mosaic 04

A negative minor can become a positive cofactor

Use the first row and second column of the example matrix.

Source matrix
A=[2−1304512−2]

Delete row one and column two.

Minor then sign
M12=det([051−2])=−5
C12=−(−5)=5

The position has a negative checkerboard sign, so the negative minor becomes positive.

Mosaic 05

Different positions can use the same steps but different signs

Compare one negative-sign position with one positive-sign position in the same matrix.

Second row, third column

M23=det([2−112])=5
C23=−5

Third row, third column

M33=det([2−104])=8
C33=8
Mosaic 06

A zero matrix entry does not force its cofactor to be zero

The cofactor is calculated from the matrix that remains after deletion, so the value of the selected entry is not part of its own minor.

Selected zero entry

a21=0
C21=4

The cofactor can still be nonzero.

Expansion contribution

a21C21=0·4=0

The cofactor is not zero, but its expansion term is zero because the selected entry multiplies it.

Mosaic 07

A reliable four-step process separates location, arithmetic, and sign

Do not try to perform all decisions mentally at once.

01Locate the positionIdentify the requested row and column and mark their intersection.
02Delete the two linesRemove the entire selected row and column while keeping the remaining order.
03Calculate the minorEvaluate the smaller determinant carefully, using parentheses around negative entries.
04Apply the signUse the checkerboard pattern or determine whether the row-plus-column total is even or odd.
Mosaic 08

A cofactor matrix places each signed minor back into its matching position

Calculate all nine cofactors and arrange them using the same row and column coordinates as the original entries.

signed minor tilesmatching positions

Position labels must be preserved

The cofactor from the first row and second column returns to the first row and second column of the cofactor matrix. Do not transpose while building this matrix.

Mosaic 09

A complete cofactor matrix is a checkerboard of signed two-by-two determinants

For a numerical matrix, work one position at a time and use the sign pattern as a final audit.

Original matrix
B=[120345067]
Cofactor matrix
C=[−2−2118−147−610−5−2]

Each entry is a signed minor, not an entry copied from the original matrix.

Mosaic 10

Cofactors become useful building blocks in determinant expansion

Choose one complete row or column. Multiply each matrix entry on that line by its matching cofactor, then add the products.

det(A)=a11C11+a12C12+a13C13

First-row cofactors

C11=−18,C12=5,C13=−4

Expansion result

det(A)=2(−18)+(−1)(5)+3(−4)=−53
Mosaic 11

A cofactor may not contain the entry from its own position

Deleting the selected row and column removes the selected entry itself. This can make a symbolic cofactor simpler than expected.

Variable entry
P=[1x234051−1]
Cofactor of that position
M12=det([305−1])=−3
C12=3

The result does not depend on x because its row and column were deleted.

Mosaic 12

Quick checks catch most cofactor errors

Use position, dimensions, sign, and independence checks before accepting an answer.

Position check

Did you delete the requested row and column?

Size check

A three-by-three source must leave a two-by-two minor.

Sign check

Does the position say keep or reverse?

Independence check

The selected entry itself should not appear inside its own minor.

Mosaic 13

Skills Covered

These medium-level high school Advanced Algebra questions emphasize accurate organization and sign control.

Entry location

Read row and column subscripts in the correct order.

Minor construction

Delete the correct lines and keep the surviving entries aligned.

Two-by-two determinants

Evaluate the smaller determinant with correct subtraction and signs.

Cofactor signs

Use the checkerboard pattern or row-plus-column parity.

Cofactor matrices

Return every signed minor to its matching position.

Expansion connection

Multiply selected entries by their cofactors and add the contributions.

Mosaic 14

How to Approach the Test

Use the same four-stage routine for numerical entries, negative values, zero entries, and simple variables.

01Circle the requested positionRead the row number first and column number second.
02Cross out its row and columnCopy the remaining smaller matrix without changing entry order.
03Find the minorCalculate the smaller determinant and keep negative values in parentheses.
04Attach the position signKeep the minor for an even coordinate sum and reverse it for an odd coordinate sum.
05Check what was requestedDistinguish the minor, the cofactor, an expansion term, and the complete determinant.
Mosaic 15

Common Mistakes

Most wrong answers result from confusing a location, a minor, a cofactor, or an expansion contribution.

Reading the subscripts backward

The first subscript is the row and the second is the column.

Deleting only the chosen entry

Remove the entire row and entire column through that position.

Reordering surviving entries

Keep the original left-to-right and top-to-bottom order in the minor.

Calling the minor a cofactor

Apply the checkerboard sign before reporting a cofactor.

Beginning every row with KEEP

The second row begins with a sign reversal; the checkerboard continues across the whole matrix.

Combining two negative signs incorrectly

A negative minor at a FLIP position becomes a positive cofactor.

Assuming a zero entry has a zero cofactor

The cofactor comes from the remaining entries and can be nonzero.

Multiplying by the entry too early

Find the requested cofactor first; multiply by the matrix entry only for an expansion term.

Transposing the cofactor matrix

Place each cofactor in its original matching position when the question asks only for the cofactor matrix.

Final cofactor audit

Check the complete route from requested position to signed result.

Locate → Delete → Calculate → Sign
1
Were row and column coordinates read in the correct order?The first index names the row; the second names the column.
2
Were both complete lines removed?The selected row and selected column must disappear.
3
Was the smaller determinant evaluated correctly?Keep the two diagonal products and negative entries visible.
4
Was the checkerboard sign applied once?Use even or odd row-plus-column parity as confirmation.
5
Is the answer a minor, cofactor, or expansion term?These related quantities are not interchangeable.
6
Was the result placed in the correct matrix position?This matters when building an entire cofactor matrix.
Use this free 20-question practice test for high school Advanced 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.