Examine the consistency of the following system of equations: $3x - y - 2z = 2$ $2y - z = -1$ $3x - 5y = 3$

Examine the consistency of the following system of – Class 12 Mathematics

Exercise 4.5 · Q5

3x - y - 2z = 2

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Concept understanding — Matrix Equation Solving

Solving a System of Equations by the Matrix Method

A system of linear equations can be written as a single matrix equation and solved in one clean step using the inverse of a matrix. This is the Class-12 "matrix method" for simultaneous equations.

Writing the system as $AX=B$

Take the system

a_1x+b_1y+c_1z=d_1,\quad a_2x+b_2y+c_2z=d_2,\quad a_3x+b_3y+c_3z=d_3.

Collect the coefficients, the unknowns, and the constants into matrices:

A=\begin{pmatrix}a_1&b_1&c_1\\a_2&b_2&c_2\\a_3&b_3&c_3\end{pmatrix},\quad X=\begin{pmatrix}x\\y\\z\end{pmatrix},\quad B=\begin{pmatrix}d_1\\d_2\\d_3\end{pmatrix}.

Then the whole system is just

AX=B.

Solving when $A$ is invertible

If $\det(A)\neq0$ , then $A^{-1}$ exists, and multiplying both sides on the left by $A^{-1}$ gives

X=A^{-1}B,\qquad\text{where }\;A^{-1}=\frac{1}{\det(A)}\operatorname{adj}(A).

So you compute $\det(A)$ , then $\operatorname{adj}(A)$ , form $A^{-1}$ , and multiply by $B$ . The single column $X=A^{-1}B$ hands you $x$ , $y$ , $z$ at once, and because $A^{-1}$ is unique, the solution is unique.

Watch out

Multiply in the correct order: $X=A^{-1}B$ , not $BA^{-1}$ . Matrix multiplication is not commutative, and $BA^{-1}$ is not even defined here.

When $\det(A)=0$

If $\det(A)=0$ , $A^{-1}$ does not exist and the inverse method fails. The system is then either inconsistent (no solution) or has infinitely many solutions. Decide which by computing $(\operatorname{adj}A)\,B$ :

$(\operatorname{adj}A)\,B\neq O$ → no solution (inconsistent).
$(\operatorname{adj}A)\,B=O$ → infinitely many solutions (consistent, dependent). …

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The coefficient determinant is $0$ , and the equations force $4y-2z=-1$ and $4y-2z=-2$ at once — a contradiction, so the system has no solution.

Write the equations in standard form, filling in the missing terms:

\begin{cases} 3x-y-2z=2 \\ 0x+2y-z=-1 \\ 3x-5y+0z=3 \end{cases}

1. Determinant test

A=\begin{bmatrix} 3 & -1 & -2 \\ 0 & 2 & -1 \\ 3 & -5 & 0 \end{bmatrix},\qquad \det A = 3\begin{vmatrix} 2 & -1 \\ -5 & 0 \end{vmatrix} - (-1)\begin{vmatrix} 0 & -1 \\ 3 & 0 \end{vmatrix} + (-2)\begin{vmatrix} 0 & 2 \\ 3 & -5 \end{vmatrix}.

= 3(0-5) + 1(0+3) - 2(0-6) = -15+3+12 = 0.

A zero determinant means $A^{-1}$ does not exist, so the system has either no solution or infinitely many — we must test which.

2. Check consistency by elimination

Subtract eq3 from eq1 to remove $x$ :

$(3x-y-2z)-(3x-5y) = 2-3 \;\Rightarrow\; 4y-2z=-1.$ …

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Q.Examine the consistency of the following system of equations: $3x - y - 2z = 2$ $2y - z = -1$ $3x - 5y = 3$

Concept understanding — Matrix Equation Solving

Solving a System of Equations by the Matrix Method

Writing the system as $AX=B$

Solving when $A$ is invertible

When $\det(A)=0$

1. Determinant test

2. Check consistency by elimination

Method: When the Determinant is Zero — Distinguishing 'No Solution' from 'Infinitely Many'

Steps

Common Mistakes

Q.Examine the consistency of the following system of equations: 3x−y−2z=23x - y - 2z = 23x−y−2z=2 2y−z=−12y - z = -12y−z=−1 3x−5y=33x - 5y = 33x−5y=3

1. Determinant test

2. Check consistency by elimination

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Q.Examine the consistency of the following system of equations: $3x - y - 2z = 2$ $2y - z = -1$ $3x - 5y = 3$

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