Algebra Practice

Determinants and Invertibility Practice Test

Advanced Algebra Practice Test: ACT math skills.

Determinants and Invertibility Practice Test

This test has 20 questions

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

The determinant is an invertibility threshold.

This free Determinants and Invertibility 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 calculating determinants, deciding whether square matrices are invertible, recognizing singular matrices, using the two-by-two inverse formula, interpreting row relationships, finding parameter restrictions, connecting invertibility to systems, and checking inverse products. Each question has four answer choices, one correct answer, and a detailed explanation that links the determinant result to the correct structural conclusion.

Threshold verdict
Nonzero determinantInverse exists and the transformation can be undone.
Pass
Zero determinantNo inverse exists because information has collapsed.
Stop
Calculate
Compare with zero
Interpret
Verify
Audit
Map 01

One determinant test separates the two cases

For a square matrix, invertibility depends on whether its determinant crosses the zero threshold.

Core rule

Invertible

det(A)≠0⇒A−1 exists

The matrix has a unique operation that reverses its effect.

Compare with zero

Singular

det(A)=0⇒A−1 does not exist

The matrix cannot be reversed by another matrix.

Map 02

Check shape before calculating

Only square matrices have determinants, and only square matrices can have two-sided inverses in this setting.

First filter

Square matrix

The row count equals the column count, so a determinant test is available.

Rectangular matrix

The determinant is not defined by the school-level square-matrix rules.

Do not guess from entries

A matrix can contain many nonzero entries and still have determinant zero.

Map 03

The two-by-two determinant provides the fastest threshold calculation

Multiply the main diagonal, multiply the cross diagonal, and subtract in that order.

Calculation
A=[abcd],det(A)=ad−bc

Protect negative entries

Keep parentheses around negative factors until both diagonal products are complete.

Interpret after simplifying

Invertibility depends on the final determinant, not an intermediate product.

Map 04

The decision path has only one numerical checkpoint

Calculation comes first; the existence conclusion comes immediately afterward.

Decision map
calculate thedeterminantzero?inverseexistsno inverseexistsnoyes

No second condition is needed

Once a square matrix has a nonzero determinant, it is invertible. Once its determinant is zero, it is singular.

Map 05

Worked example: a determinant of one gives a particularly simple inverse

Start with the determinant verdict, then use the inverse formula.

Invertible case
Matrix and determinant
A=[3121]
det(A)=3(1)−1(2)=1

The nonzero result guarantees an inverse.

Inverse
A−1=[1−1−23]

Because the determinant equals one, no fraction remains in front.

Map 06

The inverse formula itself shows why zero is forbidden

The determinant appears in the denominator, so a zero determinant would require division by zero.

Formula structure
A−1=1ad−bc[d−b−ca],ad−bc≠0

Swap

Exchange the two main-diagonal entries.

Reverse signs

Change the signs of the off-diagonal entries.

Scale

Multiply by the reciprocal of the determinant.

Check

Multiply to recover the identity matrix.

Map 07

The identity product confirms the inverse

An inverse must undo the original matrix multiplication in either order.

Verification
[3121][1−1−23]=[1001]
A correct-looking formula result is not enough if an entry was copied incorrectly. The identity product is the strongest direct check.
Map 08

Invertibility means a complete round trip is possible

The original matrix changes an input; its inverse returns the output to that same input.

Meaning
inputmatrixactioninverseactionoriginal state

The inverse must recover every input

If distinct inputs collapse to the same output, no single reverse action can tell them apart.

Map 09

Worked example: proportional rows force the determinant to zero

One row repeats the direction of the other, so the matrix does not contain two independent row directions.

Singular case
Matrix
B=[2412]

The first row is twice the second row.

Threshold calculation
det(B)=2(2)−4(1)=0

The matrix is singular, so an inverse does not exist.

Map 10

Area gives a visual reason for the zero threshold

In two dimensions, the absolute determinant is the area scale factor. Nonzero area preserves a two-dimensional shape; zero area collapses it.

Geometry
nonzero area remainsarea collapses to zero

Collapse cannot be reversed

A line does not retain enough information to reconstruct every point of the original two-dimensional shape.

Map 11

A parameter restriction comes from setting the determinant away from zero

Calculate the determinant as an expression, locate the value that makes it zero, and exclude that value.

Parameter
Parameter matrix
K=[k326]
Invertibility condition
det(K)=6k−6=6(k−1)
K is invertible when k≠1
Map 12

A triangular matrix reveals its determinant on the main diagonal

Multiply the diagonal entries. The matrix is invertible exactly when none of those diagonal entries makes the product zero.

Fast check
T=[24−10−15004],det(T)=2(−1)(4)=−8
The entries above the diagonal do not enter this shortcut. The nonzero diagonal product proves that the matrix is invertible.
Map 13

Simple row patterns often predict the zero status before full arithmetic

These observations are useful checks, but the final determinant calculation should still support the verdict.

Structural clues

Repeated rows

Equal rows force the determinant to zero, so the matrix is singular.

Proportional rows

If one row is a multiple of another, the determinant is zero.

Row swap

Swapping two rows reverses the determinant sign but does not change whether it is zero.

Nonzero row scaling

Scaling one row by a nonzero number scales the determinant without making a nonzero value become zero.

Zero row

A complete row of zeros forces a zero determinant.

Identity matrix

det(I)=1

The identity matrix is invertible.

Map 14

Determinant properties preserve the invertibility verdict

The exact determinant may change, while its zero or nonzero status carries the key conclusion.

Useful checks

Product

det(AB)=det(A)det(B)

A product is invertible when both square factors are invertible.

Transpose

det(AT)=det(A)

Transposing does not change the determinant or the invertibility verdict.

Quick product example
det(A)=−2,det(B)=5⇒det(AB)=−10

The product determinant is nonzero, so the product matrix is invertible.

Map 15

For a coefficient matrix, invertibility means one unique system solution

A nonzero determinant allows the matrix equation to be reversed. A zero determinant cannot produce a unique solution for every constants column.

Systems

Unique solution

[211−1],D=−3

The nonzero determinant guarantees exactly one ordered-pair solution.

No unique solution

[2412],D=0

Depending on the constants, the system may have no solution or infinitely many solutions.

Map 16

Skills Covered

These medium-level high school Advanced Algebra questions test accurate calculation and correct interpretation of the zero threshold.

Coverage

Square-matrix recognition

Identify when determinant and inverse questions are defined.

Determinant calculation

Evaluate two-by-two and simple triangular determinants.

Singular detection

Connect determinant zero with the absence of an inverse.

Inverse construction

Use the two-by-two inverse formula only when allowed.

Parameter restrictions

Exclude values that make a determinant zero.

Verification

Check an inverse by multiplying to obtain the identity matrix.

Map 17

How to Approach the Test

Separate the arithmetic result from the structural verdict so that the requested answer is clear.

Method
01Check the shapeConfirm that the matrix is square before using a determinant criterion.
02Choose the shortcutUse the two-by-two rule, diagonal product, or stated determinant property.
03Calculate carefullyProtect signs and simplify to the final determinant value.
04Compare with zeroNonzero means invertible; zero means singular.
05Answer the exact questionGive the inverse, parameter restriction, system conclusion, or property requested.
Map 18

Common Mistakes

Most wrong choices use a familiar calculation but attach the opposite invertibility conclusion.

Avoid
Testing a rectangular matrix

The school-level determinant criterion applies to square matrices.

Adding diagonal products

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

Stopping at an intermediate zero

Only the fully simplified determinant controls the verdict.

Reversing the conclusion

Nonzero means invertible; zero means singular.

Using the inverse formula anyway

A zero determinant creates division by zero.

Forgetting the determinant factor

The swapped-and-signed matrix must still be scaled by the reciprocal determinant.

Excluding every negative determinant

Negative determinants are nonzero and therefore allow inverses.

Assuming zero entries mean singular

Zero entries are allowed; the complete determinant is what matters.

Skipping the identity check

Multiplication can reveal a copied sign or misplaced entry.

Map 19

Final threshold audit

Move from matrix shape to determinant value to a justified invertibility verdict.

Square → Calculate → Compare → Conclude → Check
1
Is the matrix square?Confirm that the determinant test applies.
2
Was the correct determinant rule selected?Use structure before doing unnecessary arithmetic.
3
Were negative factors and subtraction handled correctly?Finish both products before comparing with zero.
4
Is the final determinant zero or nonzero?This single comparison controls the verdict.
5
Does the conclusion match the question?State invertible, singular, allowed parameter, or system behavior.
6
If an inverse was found, does its product give the identity?Use the product as a final reliability check.
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.