Answered: "The values of a for which the given..."

DofotheroU

Answered question

2020-11-12

The values of a for which the given matrix is invertible.

Answer & Explanation

Isma Jimenez

Skilled2020-11-13Added 84 answers

Approach:
In linear algebra, the determinant is a scalar value that can be computed from the elements of a square matrix and ecodes certain properties of the linear transformation described by the matrix.
Calculation:
Given is the matrix

A = [\begin{matrix} 1 & 2 & 3 & 4 & a \\ 2 & 1 & 2 & 3 & 4 \\ 3 & 2 & 1 & 2 & 3 \\ 4 & 3 & 2 & 1 & 2 \\ a & 4 & 3 & 2 & 1 \end{matrix}]

The row echelon form of given matrix after applying row transformation,

= [\begin{matrix} a & 4 & 3 & 2 & 1 \\ 2 & 1 & 2 & 3 & 4 \\ 3 & 2 & 1 & 2 & 3 \\ 4 & 3 & 2 & 1 & 2 \\ 1 & 2 & 3 & 4 & a \end{matrix}] R_{1} \leftrightarrow R_{5}

= [\begin{matrix} a & 4 & 3 & 2 & 1 \\ 0 & \frac{a - 8}{a} & \frac{2 a - 6}{a} & \frac{3 a - 4}{a} & \frac{4 a - 2}{a} \\ 3 & 2 & 1 & 2 & 3 \\ 4 & 3 & 2 & 1 & 2 \\ 1 & 2 & 3 & 4 & a \end{matrix}] R_{2} \leftarrow R_{2} - \frac{2}{a} \cdot R_{1}

= [\begin{matrix} a & 4 & 3 & 2 & 1 \\ 0 & \frac{a - 8}{a} & \frac{2 a - 6}{a} & \frac{3 a - 4}{a} & \frac{4 a - 2}{a} \\ 0 & \frac{2 a - 12}{a} & \frac{a - 9}{a} & \frac{2 a - 6}{a} & \frac{3 a - 3}{a} \\ 4 & 3 & 2 & 1 & 2 \\ 1 & 2 & 3 & 4 & a \end{matrix}] R_{3} \leftarrow R_{3} - \frac{3}{a} \cdot R_{1}

= [\begin{matrix} a & 4 & 3 & 2 & 1 \\ 0 & \frac{a - 8}{a} & \frac{2 a - 6}{a} & \frac{3 a - 4}{a} & \frac{4 a - 2}{a} \\ 0 & \frac{2 a - 12}{a} & \frac{a - 9}{a} & \frac{2 a - 6}{a} & \frac{3 a - 3}{a} \\ 0 & \frac{3 a - 16}{a} & \frac{2 a - 12}{a} & \frac{a - 8}{a} & \frac{2 a - 4}{a} \\ 1 & 2 & 3 & 4 & a \end{matrix}] R_{4} \leftarrow R_{4} - \frac{4}{a} \cdot R_{1}

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