Matrices & Determinants
Matrices and Determinants
Allen Star Batch
Grade 12
Question:
Number of distinct real values of $K$, such that the system of equations $x+2y+z=1$, $x+3y+4z=K$, $x+5y+10z=K^2$ has infinitely many solutions is:
Step-by-Step Solution
Key Concept: For a system Ax = b to have infinitely many solutions when |A| = 0, the augmented matrix must have the same rank as coefficient matrix A, requiring rank(A) = rank([A|b]) < 3. This mandates that adj(A)·b = 0, yielding constraint equations in K.
For $A = \begin{pmatrix} 1 & 2 & 1 \\ 1 & 3 & 4 \\ 1 & 5 & 10 \end{pmatrix}$, computing $|A| = 0$. Using $\text{adj}(A) \cdot B = \begin{pmatrix} 10 & -15 & 5 \\ -6 & 9 & -3 \\ 2 & -3 & 1 \end{pmatrix} \begin{pmatrix} 1 \\ K \\ K^2 \end{pmatrix}$ and solving $K^2 - 3K + 2 = 0$ yields $K = 1$ or $K = 2$.
Correct Answer: 3