Concept understanding — Solution of a System of Linear Equations by Method of Inversion
A system of n linear equations in n unknowns, such as a1x+b1y+c1z=d1, a2x+b2y+c2z=d2, a3x+b3y+c3z=d3, can be written as a single matrix equation AX=B, where A is the n×n matrix of coefficients, X is the n×1 column of unknowns, and B is the n×1 column of constants. Provided a unique solution exists, A must be non-singular, so A−1 exists. Pre-multiplying both sides of AX=B by A−1 gives A−1(AX)=A−1B, i.e. (A−1A)X=A−1B, i.e. IX=A−1B, so X=A−1B — this single matrix product delivers every unknown at once, read off as the corresponding entries of the column X. A−1 itself can be found by either the elementary-transformation method or the adjoint method; the adjoint formula A−1=∣A∣1(adjA) is usually the more direct route once the system is set up. Because the method genuinely requires A−1, it silently fails whenever ∣A∣=0 — a coefficient matrix that turns out singular means the method of inversion cannot produce a unique answer, and the equations must instead be checked directly for consistency (do they describe parallel/coincident lines or planes, giving no solution or infinitely many).
∣A∣=0 here, so A is singular and the method of inversion cannot be applied -- check consistency directly instead.
✓Final answer
A−1 does not exist; the equations are inconsistent (no solution).
Step 1: 2x+6y=8,x+3y=5 becomes [2163][xy]=[85], i.e. AX=B.
Step 2: ∣A∣=2(3)−1(6)=6−6=0. Since A is singular, A−1 does not exist, so the method of inversion cannot produce a solution for this system.
Step 3: Checking consistency directly: dividing the first equation by 2 gives x+3y=4, which contradicts the second equation x+3y=5 (the same left-hand side cannot equal two different values). So the two lines are parallel and distinct, and the system has no solution.
✓Final answer
∣A∣=0⇒A−1 does not exist; the method of inversion fails here because the two given equations are inconsistent (parallel lines, no common point).
Write as AX=B and check ∣A∣ before attempting inversion.
Trying to force a numeric answer by inverting a singular matrix (undefined -- division by ∣A∣=0)
Not checking ∣A∣ before starting the adjoint computation, and only discovering the problem partway through
Concluding "infinitely many solutions" without checking -- here the lines are parallel but NOT coincident, so there is actually no solution, not infinitely many
Q.If three numbers are added, their sum is 2. If two times the second number is subtracted from the sum of first and third numbers we get 8 and if three times the first number is added to the sum of second and third numbers we get 4. Find the numbers using matrices.
›Reveal solutionSolution
Translate the word problem into 3 linear equations, write as AX=B, and solve by elimination (equivalent to matrix reduction).
Let the numbers be x,y,z.
"If three numbers are added, their sum is 2": x+y+z=2 ... (i)
"If two times the second number is subtracted from the sum of first and third we get 8": (x+z)−2y=8⟹x−2y+z=8 ... (ii)
"If three times the first number is added to the sum of second and third we get 4": 3x+(y+z)=4⟹3x+y+z=4 ... (iii)
In matrix form AX=B: A=1131−21111, X=xyz, B=284
Q.The cost of 4 dozen pencils, 3 dozen pens and 2 dozen erasers is ₹60. The cost of 2 dozen pencils, 4 dozen pens and 6 dozen erasers is ₹90 whereas the cost of 6 dozen pencils, 2 dozen pens and 3 dozen erasers is ₹70. Find the cost of each item per dozen by using matrices.
›Reveal solutionSolution
Set up AX=B from the three cost equations and solve using Cramer's rule (determinants).
Let x,y,z = cost per dozen of pencils, pens, erasers respectively.
4x+3y+2z=60
2x+4y+6z=90
6x+2y+3z=70
In matrix form AX=B with A=426342263, X=xyz, B=609070.