2 marksMathematicsMatrices, determinants and linear systemsSolving linear systemsHardabout 5 min

GATE 2021 Question 39: Matrices, determinants and linear systems

Given below is an inter-industry transactions matrix. If final demand for the agriculture sector changes from 150 units to 300 units and for the manufacturing sector changes from 120 units to 200 units, then the output of the agriculture sector should be ______ units (in integer).

Producer \ User	Agriculture	Manufacturing	Final Demand	Total Output
Agriculture	500	350	150	1000
Manufacturing	320	360	120	800

Answer and worked solution

Answer: 1870

Step 1 - Compute technical coefficients ( $a_{ij}$ = input from sector $i$ used by sector $j$ , per unit of sector $j$ 's output):

$a_{AA} = \frac{500}{1000}=0.5, \quad a_{AM}=\frac{350}{800}=0.4375$ $a_{MA} = \frac{320}{1000}=0.32, \quad a_{MM}=\frac{360}{800}=0.45$

Step 2 - Set up the balance equations with the NEW final demand ( $FD_A=300$ , $FD_M=200$ ):

$X_A = 0.5X_A + 0.4375X_M + 300$ $X_M = 0.32X_A + 0.45X_M + 200$

Step 3 - Solve the simultaneous system. From the first equation: $0.5X_A = 0.4375X_M+300 \Rightarrow X_A = 0.875X_M+600$ .

Substituting into the second: $X_M = 0.32(0.875X_M+600)+0.45X_M+200 = 0.28X_M+192+0.45X_M+200$

$X_M - 0.73X_M = 392 \Rightarrow 0.27X_M = 392 \Rightarrow X_M \approx 1451.85$

$X_A = 0.875(1451.85)+600 \approx 1870.37$

Therefore the output of the agriculture sector should be approximately 1870 units (rounded to the nearest integer).

The trap: 2100

Students sometimes assume output should scale proportionally with the final demand change (e.g., agriculture's final demand doubled, so output might naively be assumed to roughly double too), missing that the interdependence between sectors (agriculture needs manufacturing inputs and vice versa) means the actual required output change must come from solving the full simultaneous system, not simple proportional scaling.

Matrices, determinants and linear systems has come up 102 times in 7 exams since 2006.

What this question tests

This question is on Solving linear systems, part of Matrices, determinants and linear systems in Mathematics.

Finding solutions of simultaneous linear equations by inversion, determinants or elimination.

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