**Question 1:**If $ M $ is the matrix:\[\begin{bmatrix}1 & 3 & 5 & -1 \\0 & 2 & 1 & -2 \\7 & -4 & 0 & 1 \\2 & 2 & -1 & 3 \\\end{bmatrix}\]then what is $|M|$? [Answer input box here]

M=135-1021-27-40122-13∴M→C'2=C2-2C31-75-1001-27-40124-13→C'4=C4+2C31-75900107-40124-11 Expanding…

Answered: If M is the matrix: 1 3 5 021 7 -4 0…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Step 1: Finding the determinant of the matrix

$M = (\begin{matrix} 1 & 3 & 5 & - 1 \\ 0 & 2 & 1 & - 2 \\ 7 & - 4 & 0 & 1 \\ 2 & 2 & - 1 & 3 \end{matrix}) ∴ M \overset{C'_{2} = C_{2} - 2 C_{3}}{\to} (\begin{matrix} 1 & - 7 & 5 & - 1 \\ 0 & 0 & 1 & - 2 \\ 7 & - 4 & 0 & 1 \\ 2 & 4 & - 1 & 3 \end{matrix}) \to_{}^{C'_{4} = C_{4} + 2 C_{3}} (\begin{matrix} 1 & - 7 & 5 & 9 \\ 0 & 0 & 1 & 0 \\ 7 & - 4 & 0 & 1 \\ 2 & 4 & - 1 & 1 \end{matrix})$

Expanding along row 2

$(\begin{matrix} 1 & - 7 & 5 & 9 \\ 0 & 0 & 1 & 0 \\ 7 & - 4 & 0 & 1 \\ 2 & 4 & - 1 & 1 \end{matrix}) = (0) {(- 1)}^{2 + 1} |\begin{matrix} - 7 & 5 & 9 \\ - 4 & 0 & 1 \\ 4 & - 1 & 1 \end{matrix}| + (0) {(- 1)}^{2 + 2} |\begin{matrix} 1 & 5 & 9 \\ 7 & 0 & 1 \\ 2 & - 1 & 1 \end{matrix}| + (1) {(- 1)}^{2 + 3} |\begin{matrix} 1 & - 7 & 9 \\ 7 & - 4 & 1 \\ 2 & 4 & 1 \end{matrix}| + (1) {(- 1)}^{2 + 4} |\begin{matrix} 1 & - 7 & 5 \\ 7 & - 4 & 0 \\ 2 & 4 & - 1 \end{matrix}| = - |\begin{matrix} 1 & - 7 & 9 \\ 7 & - 4 & 1 \\ 2 & 4 & 1 \end{matrix}|$

Add column $1$ multiplied by $7$ to column $2 : C_{2} = C_{2} + 7 C_{1}$

$|\begin{matrix} 1 & 0 & 9 \\ 7 & 45 & 1 \\ 2 & 18 & 1 \end{matrix}|$

Now $C'_{3} = C_{3} - 9 C_{1}$

$|\begin{matrix} 1 & 0 & 0 \\ 7 & 45 & - 62 \\ 2 & 18 & - 17 \end{matrix}|$

Expanding along row $1$

$|\begin{matrix} 1 & 0 & 0 \\ 7 & 45 & - 62 \\ 2 & 18 & - 17 \end{matrix}| = (1) {(- 1)}^{1 + 1} |\begin{matrix} 45 & - 62 \\ 18 & - 17 \end{matrix}| + (0) {(- 1)}^{1 + 2} |\begin{matrix} 7 & - 62 \\ 2 & - 17 \end{matrix}| + (0) {(- 1)}^{1 + 2} |\begin{matrix} 7 & 45 \\ 2 & 18 \end{matrix}| = |\begin{matrix} 45 & - 62 \\ 18 & - 17 \end{matrix}| = (45) \cdot (- 17) - (- 62) \cdot (18) = 351 ∴ - |\begin{matrix} 1 & - 7 & 9 \\ 7 & - 4 & 1 \\ 2 & 4 & 1 \end{matrix}| = - 351$