Determinants

Mathematics · FE Reference Handbook section

Mathematics

3 formulas

10 exam-style examples

~51 min

All Mathematics lectures

Handbook notes for this section

Definitions and conditions exactly as the handbook states them.

A determinant of order n consists of n2 numbers, called the elements of the determinant, arranged in n rows and n columns and
In any determinant, the minor of a given element is the determinant that remains after all of the elements are struck out that lie
in the same row and in the same column as the given element. Consider an element which lies in the jth column and the ith row.
The cofactor of this element is the value of the minor of the element (if i + j is even), and it is the negative of the value of the
If n is greater than 1, the value of a determinant of order n is the sum of the n products formed by multiplying each element of
some specified row (or column) by its cofactor. This sum is called the expansion of the determinant [according to the elements
of the specified row (or column)]. For a second-order determinant:

Core formulas for this FE topic

Definitions, applicability, units, assumptions and worked examples for each relation.

Worked exam-style examples

The four ways this section is written on the real exam — thoughts first, then equations, then substitution.

Example 1

Determinant of a 2x2 matrix — solve for determinant — Determinants

A student computes the determinant of a 2x2 coefficient matrix. Given a (a) = 4.5000; b (b) = 7.5000; c (c) = 6.5000; d (d) = 11.0000, determine the determinant (D).

Given

$a (a) = 4.5000$
$b (b) = 7.5000$
$c (c) = 6.5000$
$d (d) = 11.0000$

Find

determinant (D)

Start with the thinking

The governing relation printed in this handbook section is Determinant of a 2x2 matrix.
Everything except D is given, so isolate D symbolically first — never rearrange after the numbers are in.
Tabulate each given with its unit and confirm the units are consistent with the relation before substituting.
The determinant of a 2x2 matrix is computed as the product of the diagonals minus the off-diagonals.

Step-by-step solution

Step 1 — State the governing relation:
$D = a d - b c$
Step 2 — Rearrange symbolically for D:
$D = ad - bc$
Step 3
$List the givens: a (a) = 4.5000, b (b) = 7.5000, c (c) = 6.5000, d (d) = 11.0000$
Step 4 — Substitute the given values:
$D = a11.0000 - b6.5000$
Step 5 — Evaluate:
$D = 0.7500$
Step 6 — Check: returning D = 0.7500 to
$D = a d - b c$
reproduces the given quantities, and both sides carry the same units.

Answer:

D = 0.7500

Why the other options are there

1.5000 — kept a factor of two that cancels in the correct rearrangement.
0.3750 — dropped that same factor in the other direction.
0.8250 — rounded an intermediate value before the final step.

Reference: FE Handbook — Determinants

Example 2

Determinant of a 2x2 matrix — solve for a — Determinants (2)

An engineer checks whether a system's determinant is non-zero before inverting a matrix. Given b (b) = 5.0000; c (c) = 6.5000; d (d) = 12.5000; determinant (D) = 12.0000, determine the a (a).

Given

$b (b) = 5.0000$
$c (c) = 6.5000$
$d (d) = 12.5000$
$determinant (D) = 12.0000$

Find

a (a)

Start with the thinking

The governing relation printed in this handbook section is Determinant of a 2x2 matrix.
Everything except a is given, so isolate a symbolically first — never rearrange after the numbers are in.
Tabulate each given with its unit and confirm the units are consistent with the relation before substituting.
The determinant of a 2x2 matrix is computed as the product of the diagonals minus the off-diagonals.

Step-by-step solution

Step 1 — State the governing relation:
$D = a d - b c$
Step 2 — Rearrange symbolically for a:
$a = \dfrac{D + bc}{d}$
Step 3
$List the givens: b (b) = 5.0000, c (c) = 6.5000, d (d) = 12.5000, determinant (D) = 12.0000$
Step 4 — Substitute the given values:
$a = \dfrac{12.0000 + b6.5000}{12.5000}$
Step 5 — Evaluate:
$a = 3.5600$
Step 6 — Check: returning a = 3.5600 to
$D = a d - b c$
reproduces the given quantities, and both sides carry the same units.

Answer:

a = 3.5600

Why the other options are there

7.1200 — kept a factor of two that cancels in the correct rearrangement.
1.7800 — dropped that same factor in the other direction.
3.9160 — rounded an intermediate value before the final step.

Reference: FE Handbook — Determinants

Example 3

Determinant of a 2x2 matrix — solve for d — Determinants (3)

The determinant of a Jacobian matrix is evaluated at a point. Given a (a) = 5.5000; b (b) = 1.0000; c (c) = 2.5000; determinant (D) = 2.5000, determine the d (d).

Given

$a (a) = 5.5000$
$b (b) = 1.0000$
$c (c) = 2.5000$
$determinant (D) = 2.5000$

Find

d (d)

Start with the thinking

The governing relation printed in this handbook section is Determinant of a 2x2 matrix.
Everything except d is given, so isolate d symbolically first — never rearrange after the numbers are in.
Tabulate each given with its unit and confirm the units are consistent with the relation before substituting.
The determinant of a 2x2 matrix is computed as the product of the diagonals minus the off-diagonals.

Step-by-step solution

Step 1 — State the governing relation:
$D = a d - b c$
Step 2 — Rearrange symbolically for d:
$d = \dfrac{D + bc}{a}$
Step 3
$List the givens: a (a) = 5.5000, b (b) = 1.0000, c (c) = 2.5000, determinant (D) = 2.5000$
Step 4 — Substitute the given values:
$d = \dfrac{2.5000 + b2.5000}{5.5000}$
Step 5 — Evaluate:
$d = 0.9091$
Step 6 — Check: returning d = 0.9091 to
$D = a d - b c$
reproduces the given quantities, and both sides carry the same units.

Answer:

d = 0.9091

Why the other options are there

1.8182 — kept a factor of two that cancels in the correct rearrangement.
0.4545 — dropped that same factor in the other direction.
1.0000 — rounded an intermediate value before the final step.

Reference: FE Handbook — Determinants

Example 4

Determinant of a 2x2 matrix — solve for determinant (case 2) — Determinants (4)

A student computes the determinant of a 2x2 coefficient matrix. Given a (a) = 6.0000; b (b) = 5.5000; c (c) = 1.0000; d (d) = 13.0000, determine the determinant (D).

Given

$a (a) = 6.0000$
$b (b) = 5.5000$
$c (c) = 1.0000$
$d (d) = 13.0000$

Find

determinant (D)

Start with the thinking

The governing relation printed in this handbook section is Determinant of a 2x2 matrix.
Everything except D is given, so isolate D symbolically first — never rearrange after the numbers are in.
Tabulate each given with its unit and confirm the units are consistent with the relation before substituting.
The determinant of a 2x2 matrix is computed as the product of the diagonals minus the off-diagonals.

Step-by-step solution

Step 1 — State the governing relation:
$D = a d - b c$
Step 2 — Rearrange symbolically for D:
$D = ad - bc$
Step 3
$List the givens: a (a) = 6.0000, b (b) = 5.5000, c (c) = 1.0000, d (d) = 13.0000$
Step 4 — Substitute the given values:
$D = a13.0000 - b1.0000$
Step 5 — Evaluate:
$D = 72.5000$
Step 6 — Check: returning D = 72.5000 to
$D = a d - b c$
reproduces the given quantities, and both sides carry the same units.

Answer:

D = 72.5000

Why the other options are there

145.0 — kept a factor of two that cancels in the correct rearrangement.
36.2500 — dropped that same factor in the other direction.
79.7500 — rounded an intermediate value before the final step.

Reference: FE Handbook — Determinants

Example 5

Determinant of a 2x2 matrix — solve for a (case 2) — Determinants (5)

An engineer checks whether a system's determinant is non-zero before inverting a matrix. Given b (b) = 6.5000; c (c) = 3.5000; d (d) = 8.0000; determinant (D) = 68.4000, determine the a (a).

Given

$b (b) = 6.5000$
$c (c) = 3.5000$
$d (d) = 8.0000$
$determinant (D) = 68.4000$

Find

a (a)

Start with the thinking

The governing relation printed in this handbook section is Determinant of a 2x2 matrix.
Everything except a is given, so isolate a symbolically first — never rearrange after the numbers are in.
Tabulate each given with its unit and confirm the units are consistent with the relation before substituting.
The determinant of a 2x2 matrix is computed as the product of the diagonals minus the off-diagonals.

Step-by-step solution

Step 1 — State the governing relation:
$D = a d - b c$
Step 2 — Rearrange symbolically for a:
$a = \dfrac{D + bc}{d}$
Step 3
$List the givens: b (b) = 6.5000, c (c) = 3.5000, d (d) = 8.0000, determinant (D) = 68.4000$
Step 4 — Substitute the given values:
$a = \dfrac{68.4000 + b3.5000}{8.0000}$
Step 5 — Evaluate:
$a = 11.3938$
Step 6 — Check: returning a = 11.3938 to
$D = a d - b c$
reproduces the given quantities, and both sides carry the same units.

Answer:

a = 11.3938

Why the other options are there

22.7875 — kept a factor of two that cancels in the correct rearrangement.
5.6969 — dropped that same factor in the other direction.
12.5331 — rounded an intermediate value before the final step.

Reference: FE Handbook — Determinants

Example 6

Determinant of a 2x2 matrix — solve for d (case 2) — Determinants (6)

The determinant of a Jacobian matrix is evaluated at a point. Given a (a) = 8.0000; b (b) = 3.5000; c (c) = 6.0000; determinant (D) = 64.5000, determine the d (d).

Given

$a (a) = 8.0000$
$b (b) = 3.5000$
$c (c) = 6.0000$
$determinant (D) = 64.5000$

Find

d (d)

Start with the thinking

The governing relation printed in this handbook section is Determinant of a 2x2 matrix.
Everything except d is given, so isolate d symbolically first — never rearrange after the numbers are in.
Tabulate each given with its unit and confirm the units are consistent with the relation before substituting.
The determinant of a 2x2 matrix is computed as the product of the diagonals minus the off-diagonals.

Step-by-step solution

Step 1 — State the governing relation:
$D = a d - b c$
Step 2 — Rearrange symbolically for d:
$d = \dfrac{D + bc}{a}$
Step 3
$List the givens: a (a) = 8.0000, b (b) = 3.5000, c (c) = 6.0000, determinant (D) = 64.5000$
Step 4 — Substitute the given values:
$d = \dfrac{64.5000 + b6.0000}{8.0000}$
Step 5 — Evaluate:
$d = 10.6875$
Step 6 — Check: returning d = 10.6875 to
$D = a d - b c$
reproduces the given quantities, and both sides carry the same units.

Answer:

d = 10.6875

Why the other options are there

21.3750 — kept a factor of two that cancels in the correct rearrangement.
5.3438 — dropped that same factor in the other direction.
11.7563 — rounded an intermediate value before the final step.

Reference: FE Handbook — Determinants

Example 7

Determinant of a 2x2 matrix — solve for determinant (case 3) — Determinants (7)

A student computes the determinant of a 2x2 coefficient matrix. Given a (a) = 7.5000; b (b) = 7.5000; c (c) = 5.0000; d (d) = 9.0000, determine the determinant (D).

Given

$a (a) = 7.5000$
$b (b) = 7.5000$
$c (c) = 5.0000$
$d (d) = 9.0000$

Find

determinant (D)

Start with the thinking

The governing relation printed in this handbook section is Determinant of a 2x2 matrix.
Everything except D is given, so isolate D symbolically first — never rearrange after the numbers are in.
Tabulate each given with its unit and confirm the units are consistent with the relation before substituting.
The determinant of a 2x2 matrix is computed as the product of the diagonals minus the off-diagonals.

Step-by-step solution

Step 1 — State the governing relation:
$D = a d - b c$
Step 2 — Rearrange symbolically for D:
$D = ad - bc$
Step 3
$List the givens: a (a) = 7.5000, b (b) = 7.5000, c (c) = 5.0000, d (d) = 9.0000$
Step 4 — Substitute the given values:
$D = a9.0000 - b5.0000$
Step 5 — Evaluate:
$D = 30.0000$
Step 6 — Check: returning D = 30.0000 to
$D = a d - b c$
reproduces the given quantities, and both sides carry the same units.

Answer:

D = 30.0000

Why the other options are there

60.0000 — kept a factor of two that cancels in the correct rearrangement.
15.0000 — dropped that same factor in the other direction.
33.0000 — rounded an intermediate value before the final step.

Reference: FE Handbook — Determinants

Example 8

Determinant of a 2x2 matrix — solve for a (case 3) — Determinants (8)

An engineer checks whether a system's determinant is non-zero before inverting a matrix. Given b (b) = 6.0000; c (c) = 2.5000; d (d) = 10.0000; determinant (D) = 104.0, determine the a (a).

Given

$b (b) = 6.0000$
$c (c) = 2.5000$
$d (d) = 10.0000$
$determinant (D) = 104.0$

Find

a (a)

Start with the thinking

The governing relation printed in this handbook section is Determinant of a 2x2 matrix.
Everything except a is given, so isolate a symbolically first — never rearrange after the numbers are in.
Tabulate each given with its unit and confirm the units are consistent with the relation before substituting.
The determinant of a 2x2 matrix is computed as the product of the diagonals minus the off-diagonals.

Step-by-step solution

Step 1 — State the governing relation:
$D = a d - b c$
Step 2 — Rearrange symbolically for a:
$a = \dfrac{D + bc}{d}$
Step 3
$List the givens: b (b) = 6.0000, c (c) = 2.5000, d (d) = 10.0000, determinant (D) = 104.0$
Step 4 — Substitute the given values:
$a = \dfrac{104.0 + b2.5000}{10.0000}$
Step 5 — Evaluate:
$a = 11.9000$
Step 6 — Check: returning a = 11.9000 to
$D = a d - b c$
reproduces the given quantities, and both sides carry the same units.

Answer:

a = 11.9000

Why the other options are there

23.8000 — kept a factor of two that cancels in the correct rearrangement.
5.9500 — dropped that same factor in the other direction.
13.0900 — rounded an intermediate value before the final step.

Reference: FE Handbook — Determinants

Example 9

Determinant of a 2x2 matrix — solve for d (case 3) — Determinants (9)

The determinant of a Jacobian matrix is evaluated at a point. Given a (a) = 1.5000; b (b) = 3.5000; c (c) = 7.5000; determinant (D) = 145.1, determine the d (d).

Given

$a (a) = 1.5000$
$b (b) = 3.5000$
$c (c) = 7.5000$
$determinant (D) = 145.1$

Find

d (d)

Start with the thinking

The governing relation printed in this handbook section is Determinant of a 2x2 matrix.
Everything except d is given, so isolate d symbolically first — never rearrange after the numbers are in.
Tabulate each given with its unit and confirm the units are consistent with the relation before substituting.
The determinant of a 2x2 matrix is computed as the product of the diagonals minus the off-diagonals.

Step-by-step solution

Step 1 — State the governing relation:
$D = a d - b c$
Step 2 — Rearrange symbolically for d:
$d = \dfrac{D + bc}{a}$
Step 3
$List the givens: a (a) = 1.5000, b (b) = 3.5000, c (c) = 7.5000, determinant (D) = 145.1$
Step 4 — Substitute the given values:
$d = \dfrac{145.1 + b7.5000}{1.5000}$
Step 5 — Evaluate:
$d = 114.2$
Step 6 — Check: returning d = 114.2 to
$D = a d - b c$
reproduces the given quantities, and both sides carry the same units.

Answer:

d = 114.2

Why the other options are there

228.5 — kept a factor of two that cancels in the correct rearrangement.
57.1167 — dropped that same factor in the other direction.
125.7 — rounded an intermediate value before the final step.

Reference: FE Handbook — Determinants

Example 10

Determinant of a 2x2 matrix — solve for determinant (case 4) — Determinants (10)

A student computes the determinant of a 2x2 coefficient matrix. Given a (a) = 5.5000; b (b) = 5.5000; c (c) = 4.0000; d (d) = 8.0000, determine the determinant (D).

Given

$a (a) = 5.5000$
$b (b) = 5.5000$
$c (c) = 4.0000$
$d (d) = 8.0000$

Find

determinant (D)

Start with the thinking

The governing relation printed in this handbook section is Determinant of a 2x2 matrix.
Everything except D is given, so isolate D symbolically first — never rearrange after the numbers are in.
Tabulate each given with its unit and confirm the units are consistent with the relation before substituting.
The determinant of a 2x2 matrix is computed as the product of the diagonals minus the off-diagonals.

Step-by-step solution

Step 1 — State the governing relation:
$D = a d - b c$
Step 2 — Rearrange symbolically for D:
$D = ad - bc$
Step 3
$List the givens: a (a) = 5.5000, b (b) = 5.5000, c (c) = 4.0000, d (d) = 8.0000$
Step 4 — Substitute the given values:
$D = a8.0000 - b4.0000$
Step 5 — Evaluate:
$D = 22.0000$
Step 6 — Check: returning D = 22.0000 to
$D = a d - b c$
reproduces the given quantities, and both sides carry the same units.

Answer:

D = 22.0000

Why the other options are there

44.0000 — kept a factor of two that cancels in the correct rearrangement.
11.0000 — dropped that same factor in the other direction.
24.2000 — rounded an intermediate value before the final step.

Reference: FE Handbook — Determinants

Determinants

Handbook notes for this section

Core formulas for this FE topic

=a1b2−a2b1= a_{1}b_{2} - a_{2}b_{1}=a1​b2​−a2​b1​

b1b2b3=a1b2c3+a2b3c1+a3b1c2−a3b2c1−a2b1c3−a1b3c2b_{1} b_{2} b_{3} = a_{1}b_{2}c_{3} + a_{2}b_{3}c_{1} + a_{3}b_{1}c_{2} - a_{3}b_{2}c_{1} - a_{2}b_{1}c_{3} - a_{1}b_{3}c_{2}b1​b2​b3​=a1​b2​c3​+a2​b3​c1​+a3​b1​c2​−a3​b2​c1​−a2​b1​c3​−a1​b3​c2​

Worked exam-style examples