Matrices and Determinants: Solving Simultaneous Equations and Input–Output Model (Basics) Questions

Question 1

Define a matrix.

Accepted Answer

A matrix is a rectangular arrangement of numbers/symbols in rows and columns.

Question 2

What is the order of a matrix?

Accepted Answer

Order of a matrix is the number of rows × number of columns.

Question 3

State determinant of a 2×2 matrix [[a,b],[c,d]].

Accepted Answer

Determinant = ad − bc.

Question 4

When does inverse of a matrix exist?

Accepted Answer

Inverse exists only for a square matrix with non-zero determinant.

Question 5

What is an identity matrix?

Accepted Answer

Identity matrix has 1s on main diagonal and 0s elsewhere and acts as multiplicative identity.

Question 6

Write Leontief model equation.

Accepted Answer

Leontief model: X = AX + D.

Question 7

Find determinant of [[3,2],[5,4]].

Accepted Answer

|A| = (3×4) − (2×5) = 12 − 10 = 2.

Question 8

Solve by Cramer’s rule: 2x + y = 11, x + y = 7.

Accepted Answer

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.

Question 9

Find inverse of matrix [[2,1],[5,3]].

Accepted Answer

For A=[[2,1],[5,3]], |A| = 2×3 − 1×5 = 6 − 5 = 1.
A^{-1} = (1/1) [[3, −1],[−5, 2]] = [[3, −1],[−5, 2]].

Question 10

Explain why matrix multiplication is not commutative (AB ≠ BA).

Accepted Answer

Matrix multiplication depends on order. Even when both AB and BA are defined, their entries are computed from different row-column combinations, so results differ. Therefore generally AB ≠ BA.

Question 11

Write the matrix form of system: 2x+y=11 and x+y=7.

Accepted Answer

Coefficient matrix A=[[2,1],[1,1]], variable matrix X=[[x],[y]], constant matrix B=[[11],[7]].
So AX = B.

Question 12

In Leontief model, show how to obtain X from X = AX + D.

Accepted Answer

X = AX + D ⇒ X − AX = D ⇒ (I − A)X = D.
If (I − A) is invertible, X = (I − A)^{-1}D.

Question 13

Explain solving simultaneous linear equations using matrix inverse method with an example.

Accepted Answer

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]]

1) Find determinant |A| = (2×1) − (1×1) = 1, so inverse exists.
2) A^{-1} = (1/|A|) [[1, −1], [−1, 2]] = [[1, −1], [−1, 2]]
3) X = A^{-1}B:
[[x],[y]] = [[1, −1], [−1, 2]] [[11],[7]]
= [[11 − 7], [−11 + 14]]
= [[4],[3]]

So x = 4 and y = 3.

Question 14

Explain Cramer’s rule and state the formulas for x and y.

Accepted Answer

Cramer’s rule solves a system of two linear equations using determinants.

Given:
a1x + b1y = c1
a2x + b2y = c2

If D ≠ 0, the unique solution is:
x = Dx/D and y = Dy/D

This method is quick for 2 variables and is frequently asked in exams.

Question 15

Explain the Leontief input–output model and show the solution for output vector.

Accepted Answer

In Leontief input–output model, industries require inputs from each other to produce output.

Let A be the input coefficient matrix, X be total output vector, and D be final demand vector.

Total output is used for intermediate demand (AX) plus final demand (D):
X = AX + D

Rearrange:
(I − A)X = D

If (I − A) is invertible, then:
X = (I − A)^{-1} D

Interpretation: (I − A)^{-1} (Leontief inverse) shows total output required to meet final demand considering inter-industry requirements.

Matrices and Determinants: Solving Simultaneous Equations and Input–Output Model (Basics)

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