Algebra Practice

Minors and Cofactors Practice Test

Advanced Algebra Practice Test: ACT math skills.

Minors and Cofactors Practice Test

This test has 20 questions

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

The minor is the calculation. The cofactor is the signed version.

This free Minors and 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 matrix positions, deleted-row-and-column submatrices, two-by-two minor calculations, the alternating sign pattern, individual cofactors, rows of minors and cofactors, simple variables, and determinant expansion. Each question has four answer choices, one correct answer, and a detailed explanation that shows exactly where the minor value becomes a signed cofactor.

MINORDelete one row and one column, then calculate the smaller determinant.
Position sign
COFACTORKeep or reverse the minor according to its original position.
Coordinates
Minors
Signs
Cofactors
Expansion
Balance 01

Start by separating the two definitions

A minor is the determinant of a smaller matrix. A cofactor uses the same minor and may change its sign.

Minor rule

Mij=determinant after deleting row i and column j

The smaller determinant is evaluated before any position sign is applied.

Keep or reverse

Cofactor rule

Cij=(−1)i+jMij

The coordinate sum decides whether the minor keeps or reverses its sign.

Balance 02

Coordinates identify both the entry and the lines to remove

In position (i,j), the first index is the row and the second is the column.

select a coordinatesurviving submatrix

Removal is positional

The selected entry is not copied into its minor. Its entire row and column disappear, leaving a smaller square matrix.

Balance 03

The sign hinge depends on whether the coordinate sum is even or odd

An even sum keeps the minor. An odd sum reverses it. This produces the familiar checkerboard pattern.

Keep positions

i+j is even ⇒Cij=Mij

These positions begin with the upper-left corner and alternate.

Reverse positions

i+j is odd ⇒Cij=−Mij

A negative minor at one of these positions becomes a positive cofactor.

Balance 04

The full process has four school-level steps

Keep the matrix-location work separate from the two-by-two arithmetic.

locateread indexesdeleterow and columncalculatefind the minorapply signget cofactor

Do not merge the final two stages

First finish the smaller determinant. Then use the original position to decide whether that value keeps or reverses its sign.

Balance 05

One matrix can supply many minor-and-cofactor comparisons

The following examples use different positions in the same numerical matrix.

Reference matrix
A=[41−23052−16]

Use row and column labels carefully; the values alone do not identify a minor.

Balance 06

At a keep position, the minor and cofactor match

The first row and first column form an even coordinate sum.

Minor

M11=det([05−16])=5
Keep

Cofactor

C11=5

No sign reversal occurs at this position.

Balance 07

At a reverse position, the same magnitude receives the opposite sign

The first row and second column form an odd coordinate sum.

Minor

M12=det([3526])=8
Reverse

Cofactor

C12=−8

The cofactor has the same magnitude as the minor but the opposite sign.

Balance 08

A negative minor and a reverse position produce a positive cofactor

This is the sign case students most often rush.

Minor at row two, column three

M23=det([412−1])=−6

Apply the reverse sign

C23=−(−6)=6

Keep the positional negative outside parentheses until the minor sign is clear.

Balance 09

A zero entry can have a large nonzero minor and cofactor

The center entry of the reference matrix is zero, but deleting its row and column leaves four nonzero values.

Center minor

M22=det([4−226])=28
Keep

Center cofactor

C22=28

The expansion contribution would still be zero because the matrix entry multiplying this cofactor is zero.

Balance 10

A row of minors becomes a row of cofactors through the sign pattern

For the first row, the positions alternate keep, reverse, keep.

First row of minors
[M11M12M13]=[58−3]
First row of cofactors
[C11C12C13]=[5−8−3]

Only the middle position changes sign in this row.

Balance 11

The matrix of minors and matrix of cofactors differ by the sign checkerboard

Every position keeps its magnitude. Only the reverse positions change sign.

minor matrixminor valuessign boardkeep or reversecofactor matrixsigned values

Do not change positions

The cofactor in each location comes from the minor in that same location. The sign layer changes values, not coordinates.

Matrix of minors
M=[58−3428−6526−3]
Matrix of cofactors
C=[5−8−3−42865−26−3]
Balance 12

Cofactors connect the smaller calculations back to the full determinant

Expand across the first row by multiplying each entry by the cofactor in the same position.

det(A)=4(5)+1(−8)+(−2)(−3)=18

Entry factor

Use the matrix entry from the selected expansion row.

Cofactor factor

Use the signed minor from the matching location.

Final sum

Add all three entry-cofactor products to get the determinant.

Balance 13

A simple variable question still follows the same minor-to-cofactor path

Find the cofactor expression first, then solve the requested equation.

Variable matrix
P=[x203141−25]
Given cofactor value
M23=−2x−2
C23=2x+2=10⇒x=4

The odd coordinate sum reverses the minor before the equation is solved.

Balance 14

Four quick checks distinguish every related quantity

These checks are especially useful when answer choices differ only by a sign or entry factor.

Size

A three-by-three source leaves a two-by-two minor.

Position

The selected entry must disappear from its own minor.

Magnitude

A minor and its cofactor always have the same absolute value.

Requested object

Identify whether the answer should be a minor, cofactor, product, or full determinant.

Balance 15

Skills Covered

These medium-level high school Advanced Algebra questions test careful position reading, short determinant calculations, and sign reasoning.

Matrix coordinates

Read row and column subscripts in the correct order.

Minor construction

Delete the correct row and column and preserve entry order.

Minor calculation

Evaluate the remaining two-by-two determinant accurately.

Position signs

Use checkerboard signs or even-and-odd coordinate sums.

Paired matrices

Compare matrices of minors and cofactors position by position.

Expansion

Connect cofactors to the determinant through matching entry products.

Balance 16

How to Approach the Test

Pause after the minor is calculated. That pause is where the cofactor sign decision belongs.

01Read the requested symbolDetermine whether the problem asks for a minor or a cofactor.
02Locate the matrix positionUse the first subscript for the row and second for the column.
03Build the smaller matrixRemove both selected lines and keep the remaining entries aligned.
04Evaluate the minorUse the two-by-two determinant rule and protect negative entries with parentheses.
05Convert if neededApply the position sign only when the requested object is a cofactor or expansion term.
Balance 17

Common Mistakes

The most common distractors are the correct minor with the wrong cofactor sign or the correct cofactor multiplied by the wrong matrix entry.

Reading column before row

The first subscript identifies the row, not the column.

Leaving part of the selected line

Remove the entire row and entire column.

Changing the surviving order

Copy the smaller matrix exactly as it appears after deletion.

Adding the two diagonal products

A two-by-two minor uses main product minus cross product.

Reporting the minor as the cofactor

Check whether the original position requires a sign reversal.

Reversing every second value by sight

Use the actual checkerboard position, especially in the second row.

Assuming zero carries into the minor

The selected entry is deleted and does not determine its minor value.

Multiplying by the source entry too soon

That multiplication belongs to expansion, not to the cofactor definition.

Confusing the two full matrices

The matrix of cofactors includes signs; the matrix of minors does not.

Final balance audit

Decide which side of the minor-to-cofactor balance the answer belongs on.

Locate → Reduce → Evaluate → Sign
1
Was the requested matrix position identified correctly?Read row first and column second.
2
Were the selected row and column both removed?The surviving matrix must remain square.
3
Was the minor calculated before the sign decision?Keep these two stages separate.
4
Does the coordinate sum say keep or reverse?Use even for keep and odd for reverse.
5
Does the answer require an entry multiplier?Use it for expansion terms, not for an individual cofactor.
6
Was the final value placed or interpreted correctly?Preserve its position when building a matrix of minors or cofactors.
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.