Unit 7
Algebra
Class 12 Mathematics
Matrix-Based System of Linear Equations
Class 12 Mathematics – Matrix-Based System of Linear Equations Notes PDF
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This chapter develops matrix-based techniques for solving simultaneous linear equations, especially Cramer’s rule, the row-equivalent matrix method, and the inverse-matrix method, for systems involving up to three variables.
1. System of Linear Equations
An equation such as ax+by=c, where a and b are not both zero, is linear in x,y.
Two or more linear equations considered simultaneously form a system of linear equations. A solution is a set of values satisfying every equation in the system.
For example,
2x+y=7, x−y=2
is a system of two linear equations in two unknowns.
2. Matrix Form of a Linear System: AX=B
Consider
a₁x+b₁y=c₁, a₂x+b₂y=c₂.
It can be written as
a₁b₁a₂b₂ xy = c₁c₂
AX=B
where:
- A = coefficient matrix,
- X = column matrix of unknowns,
- B = constant column matrix.
3. Determinants Needed for Cramer’s Rule
3.1 Determinant of a 2×2 Matrix
For the coefficient matrix
a₁b₁a₂b₂,
its determinant is
D=a₁b₂−a₂b₁.
3.2 Determinant of a 3×3 Matrix
For D = |a₁ b₁ c₁; a₂ b₂ c₂; a₃ b₃ c₃| , expand along any row or column. Expanding along the first row:
A sign error in a determinant affects every later step in Cramer’s rule. For first-row expansion, use + − +.
4. Cramer’s Rule for Two Variables
For
a₁x+b₁y=c₁,
a₂x+b₂y=c₂,
define
If D≠0, then
Solve 2x+y=7, x−y=2.
D=2(−1)−1(1)=−3.
Dₓ=7(−1)−2(1)=−9.
Dᵧ=2(2)−1(7)=−3.
x=(−9)/(−3)=3, y=(−3)/(−3)=1.
Answer: x=3, y=1.
5. Cramer’s Rule for Three Variables
For
a₁x+b₁y+c₁z=d₁,
a₂x+b₂y+c₂z=d₂,
a₃x+b₃y+c₃z=d₃,
let
To form Dₓ, replace the x-column by the constants d₁,d₂,d₃. Similarly replace the y-column for Dᵧ and the z-column for Dz.
Solve:
x+y+z=6,
2x−y+z=3,
x+2y−z=2.
Coefficient determinant:
D= |1 1 1; 2 −1 1; 1 2 −1| =7.
Dₓ=7, Dᵧ=14, Dz=21.
x=1, y=2, z=3.
6. Row-Equivalent Matrix Method
A system is written as an augmented matrix and transformed using elementary row operations until its solution is easy to read.
- Rᵢ ↔ Rⱼ: interchange two rows.
- Rᵢ → kRᵢ, k≠0: multiply a row by a non-zero constant.
- Rᵢ → Rᵢ+kRⱼ: add a multiple of one row to another.
For
2x+y=7, x−y=2,
the augmented matrix is
2171−12.
Use row operations:
R₁ → R₁−2R₂
to get
0331−12.
Thus 3y=3, so y=1, and x−1=2, so x=3.
Always write the row operation beside the step. Do not jump between matrices without showing how the new row was obtained.
7. Inverse Matrix Method
If
AX=B
and A is invertible, multiply by A⁻¹ from the left:
A⁻¹AX=A⁻¹B.
Since A⁻¹A=I,
7.1 Inverse of a 2×2 Matrix
If
A=abcd,
then, provided ad−bc≠0,
Solve 2x+y=7, x−y=2 using the inverse method.
A=211−1, X=xy, B=72.
det(A)=2(−1)−1(1)=−3.
A⁻¹=(-1/3) −1−1−12.
X=A⁻¹B= 31.
Answer: x=3, y=1.
8. Nature of Solutions and What the Determinant Tells Us
For a square system AX=B, the determinant of the coefficient matrix is especially important.
| Condition | Meaning | Consequence |
|---|---|---|
| det(A)≠0 | A is invertible | Exactly one solution; Cramer and inverse methods both apply. |
| det(A)=0 | A is singular | No inverse exists; the system may have no solution or infinitely many solutions. |
In row reduction, contradictory rows indicate no solution, while dependent rows may leave one or more free variables and hence infinitely many solutions.
9. Worked Examples
Solve 3x+2y=12, x−y=1.
D=3(−1)−2(1)=−5,
Dₓ=12(−1)−2(1)=−14,
Dᵧ=3(1)−12(1)=−9.
x=14/5, y=9/5.
Solve x+2y=5, 3x+4y=11.
1253411
R₂→R₂−3R₁
1250−2−4.
Hence y=2, then x=1.
Solve x+y=5, 2x−y=1.
A=112−1, B=51.
det(A)=−3.
A⁻¹=(-1/3) −1−1−21.
X=A⁻¹B=(2,3)T.
Answer: x=2, y=3.
Solve:
x+y+z=6,
x−y+z=2,
2x+y−z=1.
Subtract row 1 from row 2:
−2y=−4 ⇒ y=2.
Using x+z=4 and the third equation 2x−z=−1:
3x=3 ⇒ x=1, z=3.
Answer: x=1, y=2, z=3.
Consider x+y=2 and 2x+2y=4.
The second equation is exactly twice the first. Row reduction gives a zero row:
112000.
There is one free variable, so the system has infinitely many solutions.
Consider x+y=2 and 2x+2y=5.
Eliminating gives 0=1, which is impossible.
Therefore the system has no solution.
Check whether (x,y)=(2,1) satisfies 3x−y=5 and x+2y=4.
3(2)−1=5, 2+2(1)=4.
Both are true, so (2,1) is the solution.
10. Application to Word Problems
Linear systems often arise when several unknown quantities satisfy several independent conditions. The main challenge is translating the wording into equations.
Adult tickets cost NPR 200 and student tickets cost NPR 100. A total of 30 tickets produces NPR 4,500. Find the number of each type.
Let x = adult tickets, y = student tickets.
x+y=30,
200x+100y=4500.
Divide the second equation by 100:
2x+y=45.
Subtract the first equation:
x=15.
Then y=15.
Answer: 15 adult tickets and 15 student tickets.
- Define each unknown clearly.
- Translate each independent condition into one linear equation.
- Write the coefficient matrix in the same variable order in every row.
- Solve by the method requested in the question.
- Interpret the numerical answer in the original context.
11. Choosing the Best Method
| Method | Best use | Main requirement / caution |
|---|---|---|
| Cramer’s rule | Small square systems, especially 2×2 or 3×3; exam questions asking determinant method | For the direct unique-solution formula, D≠0. |
| Row-equivalent method | General systematic solving and identifying no/infinite solutions | Keep row operations accurate and preserve the augmented column. |
| Inverse matrix method | Square system written as AX=B | Requires A⁻¹, so det(A)≠0. |
12. Common Mistakes and Warnings
If columns are ordered x,y,z, every equation must follow that same order.
For Dₓ replace only the x-column; for Dᵧ only the y-column, etc.
Cramer’s direct formula cannot be used when the main determinant is zero.
When expanding across the first row, the cofactor signs are +, −, +.
In an augmented matrix, apply the row operation to the entire row, including constants.
From AX=B, multiply on the left by A⁻¹: A⁻¹AX=A⁻¹B.
A singular matrix has no ordinary inverse.
Substitute the final x,y,z into the original equations to catch arithmetic errors.
13. Exam-Important Formula Sheet
14. Important Exam Questions
Short-Answer Questions
- Write a given system of linear equations in matrix form AX=B.
- Define the coefficient matrix and augmented matrix.
- State Cramer’s rule for two variables.
- State Cramer’s rule for three variables.
- Write the three elementary row operations.
- State the inverse-matrix method for solving AX=B.
- What condition is required for A⁻¹ to exist?
- What does det(A)≠0 imply for a square linear system?
Derivation / Explanation Questions
- Derive the inverse-matrix solution X=A⁻¹B from AX=B.
- Explain how the determinants D,Dₓ,Dᵧ are formed in Cramer’s rule.
- Explain how row operations preserve an equivalent linear system.
- Describe how row reduction identifies a contradictory or dependent system.
Numerical Questions
- Solve a 2×2 system by Cramer’s rule.
- Solve a 3×3 system by Cramer’s rule.
- Solve a system using the row-equivalent matrix method.
- Solve a 2×2 or 3×3 system using the inverse-matrix method.
- Determine whether a system has a unique solution, no solution or infinitely many solutions.
- Translate a simple word problem into simultaneous linear equations and solve it by a matrix method.
Diagram / Matrix Practice
- Show the conversion of simultaneous equations into AX=B.
- Draw the column-replacement pattern used in Cramer’s rule.
- Write the row-reduction flow from an augmented matrix to echelon form.
- Illustrate geometrically the one-solution, no-solution and infinitely-many-solution cases for two variables.
15. One-Minute Revision
- A linear system can be represented compactly as AX=B.
- A contains coefficients, X unknowns, and B constants.
- For a 2×2 matrix, determinant = ad−bc.
- Cramer’s rule replaces one coefficient column at a time by the constant column.
- If D≠0, a square system has a unique solution.
- For two variables, x=Dₓ/D, y=Dᵧ/D.
- For three variables, also use z=D_z/D.
- Row operations are row interchange, non-zero row scaling and row replacement.
- Always apply a row operation to the constant column too.
- The inverse method is X=A⁻¹B.
- The inverse method requires det(A)≠0.
- A row equivalent to 0=nonzero signals no solution.
- A zero row with free variables may signal infinitely many solutions.
- Keep the variable order consistent in every equation and matrix.
- Substitute the answer back into the original equations for a final check.
16. Diagram Practice
- Two intersecting straight lines representing a unique 2×2 solution.
- Equation-to-AX=B conversion diagram.
- 3×3 determinant expansion with +, −, + signs.
- Cramer column-replacement diagram for two variables.
- Cramer column-replacement diagram for three variables.
- Row-reduction flowchart.
- Inverse-matrix method flowchart.
- Unique/no/infinite solution geometry for two equations.
Also Visit
Original Nepal eNotes page: Class 12 Mathematics Matrix-Based System of Linear Equations Notes.
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