If the system of equations 3x + y + 4z = 3, 2x + αy − z = −3, x + 2y + z = 4 has no solution, then the value of α is equal to

For a system $AX = B$ of 3 equations in 3 unknowns, define:

$D$ = determinant of coefficient matrix $A$
$D_1$ = det obtained by replacing column 1 of $A$ with $B$
$D_2$ = det obtained by replacing column 2 of $A$ with $B$
$D_3$ = det obtained by replacing column 3 of $A$ with $B$

Condition	Type of Solution
$D \neq 0$	Unique solution: $x=D_1/D$, $y=D_2/D$, $z=D_3/D$
$D=0$, all $D_i=0$	Infinite solutions (consistent)
$D=0$, any $D_i\neq 0$	No solution (inconsistent)

For matrix $\begin{pmatrix}a_1&b_1&c_1\\a_2&b_2&c_2\\a_3&b_3&c_3\end{pmatrix}$: $$D = a_1(b_2c_3-b_3c_2) - b_1(a_2c_3-a_3c_2) + c_1(a_2b_3-a_3b_2)$$ Signs follow the pattern: $+\ -\ +$ for Row 1 expansion.

Sarrus Rule (shortcut for 3×3): Write the first two columns again to the right of the matrix, then:
Sum of products along 3 main diagonals (top-left to bottom-right) MINUS sum of products along 3 anti-diagonals (top-right to bottom-left).

The rank of a matrix is the maximum number of linearly independent rows/columns.

For system $AX=B$ (augmented matrix $[A|B]$):
• Unique solution: $\rho(A) = \rho([A|B]) = 3$ (= number of unknowns)
• Infinite solutions: $\rho(A) = \rho([A|B]) < 3$
• No solution: $\rho(A) < \rho([A|B])$

When $D=0$ ($\alpha=19$), $\rho(A) \leq 2$. Since $D_1 \neq 0$, we get $\rho([A|B]) = 3 > \rho(A)$ → no solution.

With $\alpha=19$, perform row operations on augmented matrix $[A|B]$: $$\begin{pmatrix}3&1&4&|&3\\2&19&-1&|&-3\\1&2&1&|&4\end{pmatrix}$$ $R_1 \leftrightarrow R_3$: $$\begin{pmatrix}1&2&1&|&4\\2&19&-1&|&-3\\3&1&4&|&3\end{pmatrix}$$ $R_2 \to R_2-2R_1$, $R_3 \to R_3-3R_1$: $$\begin{pmatrix}1&2&1&|&4\\0&15&-3&|&-11\\0&-5&1&|&-9\end{pmatrix}$$ $R_3 \to R_3 + R_2/3$: $$\begin{pmatrix}1&2&1&|&4\\0&15&-3&|&-11\\0&0&0&|&-38/3\end{pmatrix}$$ Last row: $0 = -38/3 \neq 0$ → inconsistent → no solution ✓

Each equation $ax+by+cz=d$ represents a plane in 3D space.

Three planes can:
• Intersect at a unique point → unique solution
• All pass through a common line → infinite solutions
• All be parallel and identical → infinite solutions
• Have no common intersection → no solution

When $\alpha=19$: the three planes have no common point — they form an inconsistent system. Geometrically, they might be arranged like the faces of a triangular prism (parallel intersection lines with no common point).

1. Row/Column swap: Changes sign of determinant.
2. Scalar multiple: $|kA| = k^n|A|$ for $n\times n$ matrix.
3. Zero row/column: Determinant = 0.
4. Two identical rows/columns: Determinant = 0.
5. Adding multiple of one row to another: Determinant unchanged.
6. Transpose: $|A^T| = |A|$.
7. Product: $|AB| = |A|\cdot|B|$.
8. Linear function in one row: Can split determinant into sum of two.

❌ Mistake 1: Setting $D=0$ and directly concluding no solution — must also verify at least one $D_i \neq 0$. If all $D_i = 0$, it's infinite solutions, not no solution.

❌ Mistake 2: Sign error in determinant expansion — Row 1 cofactor signs are $+,-,+$, not all positive.

❌ Mistake 3: Arithmetic error in $2\times2$ minors. For $\begin{vmatrix}\alpha&-1\\2&1\end{vmatrix}$: $= \alpha(1)-(-1)(2) = \alpha+2$, not $\alpha-2$.

❌ Mistake 4: Confusing $D_1$ (replace column 1) with replacing row 1. Always replace the column corresponding to the variable.

$D=-\alpha+19$ $D=0 \Rightarrow \alpha=19$ $D_1=-266\neq0$ No solution confirmed ✓ Cramer's Rule: $D=0$, $D_i\neq0$ → no solution