A matrix is a rectangular arrangement of numbers/symbols in rows and columns.
A matrix A of order m×n has:
Example (2×3 matrix): [ A = egin{bmatrix} 1 & 2 & 3 \ 4 & 5 & 6 \end{bmatrix} ]
Business use: Matrices store structured data (costs, inputs, outputs, coefficients) and help solve multiple equations efficiently.
Exam point: Identity matrix behaves like 1 in multiplication: AI = IA = A.
Possible only when matrices have the same order.
kA means multiply each element by k.
If A is m×n and B is n×p, then AB is m×p.
Rule: (AB)ij = sum of (row i of A) × (column j of B).
Important: Matrix multiplication is generally not commutative: AB ≠ BA.
Determinant is defined only for square matrices.
For A = [[a, b],[c, d]] |A| = ad − bc
For 3×3, use expansion (cofactors) or Sarrus rule (if allowed).
Key exam use: If determinant is 0, the matrix is singular and has no inverse.
A matrix A has an inverse A⁻¹ only if |A| ≠ 0.
For A = [[a, b],[c, d]] A⁻¹ = (1/(ad−bc)) × [[d, −b], [−c, a]]
Check: A × A⁻¹ = I.
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Business mathematics are mathematics used by commercial enterprises to record and manage business operations. Commercial organizations use mathematics in accounting, inventory management, marketing, sales forecasting, and financial analysis.
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A matrix is a rectangular arrangement of numbers/symbols in rows and columns.
A matrix A of order m×n has:
Example (2×3 matrix): [ A = egin{bmatrix} 1 & 2 & 3 \ 4 & 5 & 6 \end{bmatrix} ]
Business use: Matrices store structured data (costs, inputs, outputs, coefficients) and help solve multiple equations efficiently.
Exam point: Identity matrix behaves like 1 in multiplication: AI = IA = A.
Possible only when matrices have the same order.
kA means multiply each element by k.
If A is m×n and B is n×p, then AB is m×p.
Rule: (AB)ij = sum of (row i of A) × (column j of B).
Important: Matrix multiplication is generally not commutative: AB ≠ BA.
Determinant is defined only for square matrices.
For A = [[a, b],[c, d]] |A| = ad − bc
For 3×3, use expansion (cofactors) or Sarrus rule (if allowed).
Key exam use: If determinant is 0, the matrix is singular and has no inverse.
A matrix A has an inverse A⁻¹ only if |A| ≠ 0.
For A = [[a, b],[c, d]] A⁻¹ = (1/(ad−bc)) × [[d, −b], [−c, a]]
Check: A × A⁻¹ = I.
A system of linear equations can be written as: AX = B Where:
If |A| ≠ 0, then: X = A⁻¹B
2x + y = 11 x + y = 7
A = [[2,1],[1,1]], X = [[x],[y]], B = [[11],[7]] Find A⁻¹ and compute X = A⁻¹B.
For two equations: a1x + b1y = c1 a2x + b2y = c2
Determinant D = |a1 b1; a2 b2| = a1b2 − a2b1
Dx = |c1 b1; c2 b2| Dy = |a1 c1; a2 c2|
If D ≠ 0: x = Dx/D, y = Dy/D
Use in exams: Fast for 2 variables, but matrix inverse method is more general.
Input–output model studies interdependence between industries.
Let A be the input coefficient matrix where aij is input from industry i needed to produce 1 unit of output of industry j.
Let X be total output vector and D be final demand vector.
Leontief model: X = AX + D (I − A)X = D X = (I − A)⁻¹ D (if inverse exists)
Interpretation: (I − A)⁻¹ is called Leontief inverse, showing total production required to meet final demand.
From this topic
|A| = (3×4) − (2×5) = 12 − 10 = 2.
D = |2 1; 1 1| = 2×1 − 1×1 = 1. Dx = |11 1; 7 1| = 11×1 − 1×7 = 4. Dy = |2 11; 1 7| = 2×7 − 11×1 = 3. So x = Dx/D = 4 and y = Dy/D = 3.
To solve simultaneous equations using matrices, write the system as AX = B.
Example: 2x + y = 11 x + y = 7
A = [[2,1],[1,1]], X = [[x],[y]], B = [[11],[7]]
So x = 4 and y = 3.