G2 - Determinants


Example 1 πŸ”—

Show how to compute the determinant of the matrix

\[A= \left[\begin{array}{cccc} -2 & 3 & 0 & 5 \\ -6 & 1 & 6 & 1 \\ 3 & 0 & 0 & -1 \\ -5 & 6 & 4 & -3 \end{array}\right] .\]

.

Answer:

\[\operatorname{det}\ \left[\begin{array}{cccc} -2 & 3 & 0 & 5 \\ -6 & 1 & 6 & 1 \\ 3 & 0 & 0 & -1 \\ -5 & 6 & 4 & -3 \end{array}\right] = -632 \]


Example 2 πŸ”—

Show how to compute the determinant of the matrix

\[A= \left[\begin{array}{cccc} -1 & 1 & 2 & -1 \\ 4 & -1 & -3 & 6 \\ 0 & 0 & 1 & -2 \\ -3 & -3 & 5 & 1 \end{array}\right] .\]

.

Answer:

\[\operatorname{det}\ \left[\begin{array}{cccc} -1 & 1 & 2 & -1 \\ 4 & -1 & -3 & 6 \\ 0 & 0 & 1 & -2 \\ -3 & -3 & 5 & 1 \end{array}\right] = -78 \]


Example 3 πŸ”—

Show how to compute the determinant of the matrix

\[A= \left[\begin{array}{cccc} -2 & 6 & 2 & -5 \\ -1 & -6 & 0 & -3 \\ -5 & 5 & 0 & -3 \\ -4 & 0 & 1 & 3 \end{array}\right] .\]

.

Answer:

\[\operatorname{det}\ \left[\begin{array}{cccc} -2 & 6 & 2 & -5 \\ -1 & -6 & 0 & -3 \\ -5 & 5 & 0 & -3 \\ -4 & 0 & 1 & 3 \end{array}\right] = -655 \]


Example 4 πŸ”—

Show how to compute the determinant of the matrix

\[A= \left[\begin{array}{cccc} -3 & 4 & 2 & -2 \\ 6 & -6 & -2 & 0 \\ 0 & 0 & 3 & -1 \\ -6 & 3 & 6 & 4 \end{array}\right] .\]

.

Answer:

\[\operatorname{det}\ \left[\begin{array}{cccc} -3 & 4 & 2 & -2 \\ 6 & -6 & -2 & 0 \\ 0 & 0 & 3 & -1 \\ -6 & 3 & 6 & 4 \end{array}\right] = -6 \]


Example 5 πŸ”—

Show how to compute the determinant of the matrix

\[A= \left[\begin{array}{cccc} 2 & 6 & -5 & 1 \\ 0 & -2 & -4 & 0 \\ 3 & 6 & 2 & -3 \\ 1 & -4 & -1 & 0 \end{array}\right] .\]

.

Answer:

\[\operatorname{det}\ \left[\begin{array}{cccc} 2 & 6 & -5 & 1 \\ 0 & -2 & -4 & 0 \\ 3 & 6 & 2 & -3 \\ 1 & -4 & -1 & 0 \end{array}\right] = -248 \]


Example 6 πŸ”—

Show how to compute the determinant of the matrix

\[A= \left[\begin{array}{cccc} 2 & 5 & 0 & -4 \\ 3 & -3 & 6 & 4 \\ 4 & 3 & -5 & -6 \\ -1 & 0 & 0 & -3 \end{array}\right] .\]

.

Answer:

\[\operatorname{det}\ \left[\begin{array}{cccc} 2 & 5 & 0 & -4 \\ 3 & -3 & 6 & 4 \\ 4 & 3 & -5 & -6 \\ -1 & 0 & 0 & -3 \end{array}\right] = -635 \]


Example 7 πŸ”—

Show how to compute the determinant of the matrix

\[A= \left[\begin{array}{cccc} -3 & -2 & -4 & 0 \\ 1 & 0 & 3 & 0 \\ -2 & -4 & -1 & 1 \\ 4 & 5 & 5 & 4 \end{array}\right] .\]

.

Answer:

\[\operatorname{det}\ \left[\begin{array}{cccc} -3 & -2 & -4 & 0 \\ 1 & 0 & 3 & 0 \\ -2 & -4 & -1 & 1 \\ 4 & 5 & 5 & 4 \end{array}\right] = -51 \]


Example 8 πŸ”—

Show how to compute the determinant of the matrix

\[A= \left[\begin{array}{cccc} 6 & 0 & 1 & -1 \\ 4 & -1 & -6 & 0 \\ -4 & -2 & -5 & 3 \\ 3 & -4 & -3 & 0 \end{array}\right] .\]

.

Answer:

\[\operatorname{det}\ \left[\begin{array}{cccc} 6 & 0 & 1 & -1 \\ 4 & -1 & -6 & 0 \\ -4 & -2 & -5 & 3 \\ 3 & -4 & -3 & 0 \end{array}\right] = 256 \]


Example 9 πŸ”—

Show how to compute the determinant of the matrix

\[A= \left[\begin{array}{cccc} -1 & -1 & 4 & 0 \\ 1 & -1 & -4 & 3 \\ 6 & 6 & 0 & 1 \\ -2 & 1 & 1 & 0 \end{array}\right] .\]

.

Answer:

\[\operatorname{det}\ \left[\begin{array}{cccc} -1 & -1 & 4 & 0 \\ 1 & -1 & -4 & 3 \\ 6 & 6 & 0 & 1 \\ -2 & 1 & 1 & 0 \end{array}\right] = 230 \]


Example 10 πŸ”—

Show how to compute the determinant of the matrix

\[A= \left[\begin{array}{cccc} 1 & -4 & -2 & 0 \\ 2 & 3 & 5 & -2 \\ -3 & -5 & 3 & 5 \\ 0 & 1 & 3 & 0 \end{array}\right] .\]

.

Answer:

\[\operatorname{det}\ \left[\begin{array}{cccc} 1 & -4 & -2 & 0 \\ 2 & 3 & 5 & -2 \\ -3 & -5 & 3 & 5 \\ 0 & 1 & 3 & 0 \end{array}\right] = -24 \]


Example 11 πŸ”—

Show how to compute the determinant of the matrix

\[A= \left[\begin{array}{cccc} -5 & -6 & -6 & -6 \\ -3 & -1 & 0 & 0 \\ 2 & 2 & 5 & 2 \\ 2 & -5 & 0 & -1 \end{array}\right] .\]

.

Answer:

\[\operatorname{det}\ \left[\begin{array}{cccc} -5 & -6 & -6 & -6 \\ -3 & -1 & 0 & 0 \\ 2 & 2 & 5 & 2 \\ 2 & -5 & 0 & -1 \end{array}\right] = -265 \]


Example 12 πŸ”—

Show how to compute the determinant of the matrix

\[A= \left[\begin{array}{cccc} -5 & 5 & -2 & -1 \\ -1 & 0 & 0 & 0 \\ 5 & 4 & 5 & 5 \\ 4 & 3 & 6 & 1 \end{array}\right] .\]

.

Answer:

\[\operatorname{det}\ \left[\begin{array}{cccc} -5 & 5 & -2 & -1 \\ -1 & 0 & 0 & 0 \\ 5 & 4 & 5 & 5 \\ 4 & 3 & 6 & 1 \end{array}\right] = -156 \]


Example 13 πŸ”—

Show how to compute the determinant of the matrix

\[A= \left[\begin{array}{cccc} 6 & -1 & -6 & -1 \\ 3 & 6 & 6 & 0 \\ -5 & 2 & -4 & 3 \\ -4 & 1 & 2 & 0 \end{array}\right] .\]

.

Answer:

\[\operatorname{det}\ \left[\begin{array}{cccc} 6 & -1 & -6 & -1 \\ 3 & 6 & 6 & 0 \\ -5 & 2 & -4 & 3 \\ -4 & 1 & 2 & 0 \end{array}\right] = 486 \]


Example 14 πŸ”—

Show how to compute the determinant of the matrix

\[A= \left[\begin{array}{cccc} 1 & -3 & -3 & -2 \\ -2 & 2 & 0 & -1 \\ 1 & -4 & 0 & -6 \\ 1 & 2 & 0 & -3 \end{array}\right] .\]

.

Answer:

\[\operatorname{det}\ \left[\begin{array}{cccc} 1 & -3 & -3 & -2 \\ -2 & 2 & 0 & -1 \\ 1 & -4 & 0 & -6 \\ 1 & 2 & 0 & -3 \end{array}\right] = 180 \]


Example 15 πŸ”—

Show how to compute the determinant of the matrix

\[A= \left[\begin{array}{cccc} 3 & 2 & 3 & 0 \\ -4 & 2 & -6 & 1 \\ 4 & -4 & 5 & -2 \\ 6 & -1 & 6 & 0 \end{array}\right] .\]

.

Answer:

\[\operatorname{det}\ \left[\begin{array}{cccc} 3 & 2 & 3 & 0 \\ -4 & 2 & -6 & 1 \\ 4 & -4 & 5 & -2 \\ 6 & -1 & 6 & 0 \end{array}\right] = -45 \]


Example 16 πŸ”—

Show how to compute the determinant of the matrix

\[A= \left[\begin{array}{cccc} 0 & -3 & -6 & 5 \\ -1 & 1 & -6 & 3 \\ 0 & 2 & 1 & 0 \\ 6 & 2 & 1 & -1 \end{array}\right] .\]

.

Answer:

\[\operatorname{det}\ \left[\begin{array}{cccc} 0 & -3 & -6 & 5 \\ -1 & 1 & -6 & 3 \\ 0 & 2 & 1 & 0 \\ 6 & 2 & 1 & -1 \end{array}\right] = -237 \]


Example 17 πŸ”—

Show how to compute the determinant of the matrix

\[A= \left[\begin{array}{cccc} -4 & -6 & 4 & 0 \\ 6 & 5 & 5 & 0 \\ 1 & 0 & 5 & -1 \\ -6 & -2 & 1 & 2 \end{array}\right] .\]

.

Answer:

\[\operatorname{det}\ \left[\begin{array}{cccc} -4 & -6 & 4 & 0 \\ 6 & 5 & 5 & 0 \\ 1 & 0 & 5 & -1 \\ -6 & -2 & 1 & 2 \end{array}\right] = 288 \]


Example 18 πŸ”—

Show how to compute the determinant of the matrix

\[A= \left[\begin{array}{cccc} 3 & 1 & 0 & 1 \\ 1 & 2 & 1 & 0 \\ -4 & -1 & -2 & 6 \\ -2 & 2 & 0 & -2 \end{array}\right] .\]

.

Answer:

\[\operatorname{det}\ \left[\begin{array}{cccc} 3 & 1 & 0 & 1 \\ 1 & 2 & 1 & 0 \\ -4 & -1 & -2 & 6 \\ -2 & 2 & 0 & -2 \end{array}\right] = 68 \]


Example 19 πŸ”—

Show how to compute the determinant of the matrix

\[A= \left[\begin{array}{cccc} -2 & 0 & -4 & 2 \\ 2 & -1 & -6 & -6 \\ -4 & 0 & -4 & 4 \\ -2 & -2 & 4 & -3 \end{array}\right] .\]

.

Answer:

\[\operatorname{det}\ \left[\begin{array}{cccc} -2 & 0 & -4 & 2 \\ 2 & -1 & -6 & -6 \\ -4 & 0 & -4 & 4 \\ -2 & -2 & 4 & -3 \end{array}\right] = 24 \]


Example 20 πŸ”—

Show how to compute the determinant of the matrix

\[A= \left[\begin{array}{cccc} 1 & 1 & -1 & 2 \\ 0 & 0 & -1 & -3 \\ 2 & -6 & -4 & 3 \\ 5 & -5 & 2 & 0 \end{array}\right] .\]

.

Answer:

\[\operatorname{det}\ \left[\begin{array}{cccc} 1 & 1 & -1 & 2 \\ 0 & 0 & -1 & -3 \\ 2 & -6 & -4 & 3 \\ 5 & -5 & 2 & 0 \end{array}\right] = 298 \]


Example 21 πŸ”—

Show how to compute the determinant of the matrix

\[A= \left[\begin{array}{cccc} -4 & 3 & 2 & 6 \\ -1 & 0 & -1 & -3 \\ 5 & 0 & 5 & -1 \\ 6 & 0 & 3 & -5 \end{array}\right] .\]

.

Answer:

\[\operatorname{det}\ \left[\begin{array}{cccc} -4 & 3 & 2 & 6 \\ -1 & 0 & -1 & -3 \\ 5 & 0 & 5 & -1 \\ 6 & 0 & 3 & -5 \end{array}\right] = -144 \]


Example 22 πŸ”—

Show how to compute the determinant of the matrix

\[A= \left[\begin{array}{cccc} -1 & 6 & 0 & 1 \\ -1 & 0 & -3 & 0 \\ 6 & -1 & -3 & -3 \\ -2 & -3 & -2 & 3 \end{array}\right] .\]

.

Answer:

\[\operatorname{det}\ \left[\begin{array}{cccc} -1 & 6 & 0 & 1 \\ -1 & 0 & -3 & 0 \\ 6 & -1 & -3 & -3 \\ -2 & -3 & -2 & 3 \end{array}\right] = -337 \]


Example 23 πŸ”—

Show how to compute the determinant of the matrix

\[A= \left[\begin{array}{cccc} -6 & -4 & 4 & -5 \\ -1 & -6 & -1 & 6 \\ 4 & -2 & 0 & 2 \\ 0 & 0 & 2 & 1 \end{array}\right] .\]

.

Answer:

\[\operatorname{det}\ \left[\begin{array}{cccc} -6 & -4 & 4 & -5 \\ -1 & -6 & -1 & 6 \\ 4 & -2 & 0 & 2 \\ 0 & 0 & 2 & 1 \end{array}\right] = 600 \]


Example 24 πŸ”—

Show how to compute the determinant of the matrix

\[A= \left[\begin{array}{cccc} -4 & -3 & 0 & -5 \\ 1 & 1 & -2 & -3 \\ -5 & -6 & 0 & 6 \\ 3 & -6 & 0 & 5 \end{array}\right] .\]

.

Answer:

\[\operatorname{det}\ \left[\begin{array}{cccc} -4 & -3 & 0 & -5 \\ 1 & 1 & -2 & -3 \\ -5 & -6 & 0 & 6 \\ 3 & -6 & 0 & 5 \end{array}\right] = -786 \]


Example 25 πŸ”—

Show how to compute the determinant of the matrix

\[A= \left[\begin{array}{cccc} -4 & 5 & 4 & -5 \\ -3 & -4 & 2 & 5 \\ -2 & 5 & 3 & 2 \\ 1 & 0 & 3 & 0 \end{array}\right] .\]

.

Answer:

\[\operatorname{det}\ \left[\begin{array}{cccc} -4 & 5 & 4 & -5 \\ -3 & -4 & 2 & 5 \\ -2 & 5 & 3 & 2 \\ 1 & 0 & 3 & 0 \end{array}\right] = -868 \]


Example 26 πŸ”—

Show how to compute the determinant of the matrix

\[A= \left[\begin{array}{cccc} 0 & 0 & 4 & -5 \\ 4 & 3 & 3 & 2 \\ -3 & 0 & -4 & 6 \\ 5 & 1 & 6 & 5 \end{array}\right] .\]

.

Answer:

\[\operatorname{det}\ \left[\begin{array}{cccc} 0 & 0 & 4 & -5 \\ 4 & 3 & 3 & 2 \\ -3 & 0 & -4 & 6 \\ 5 & 1 & 6 & 5 \end{array}\right] = 425 \]


Example 27 πŸ”—

Show how to compute the determinant of the matrix

\[A= \left[\begin{array}{cccc} -3 & 6 & -4 & 3 \\ -4 & 6 & 1 & -1 \\ 1 & 0 & -3 & 0 \\ 6 & -3 & -2 & 0 \end{array}\right] .\]

.

Answer:

\[\operatorname{det}\ \left[\begin{array}{cccc} -3 & 6 & -4 & 3 \\ -4 & 6 & 1 & -1 \\ 1 & 0 & -3 & 0 \\ 6 & -3 & -2 & 0 \end{array}\right] = 246 \]


Example 28 πŸ”—

Show how to compute the determinant of the matrix

\[A= \left[\begin{array}{cccc} 4 & 2 & -6 & -3 \\ -2 & 3 & -5 & -1 \\ -1 & -6 & 1 & -5 \\ 0 & -3 & 0 & 1 \end{array}\right] .\]

.

Answer:

\[\operatorname{det}\ \left[\begin{array}{cccc} 4 & 2 & -6 & -3 \\ -2 & 3 & -5 & -1 \\ -1 & -6 & 1 & -5 \\ 0 & -3 & 0 & 1 \end{array}\right] = -721 \]


Example 29 πŸ”—

Show how to compute the determinant of the matrix

\[A= \left[\begin{array}{cccc} -3 & -4 & 2 & -3 \\ 4 & -3 & 2 & -2 \\ 1 & 3 & 0 & 0 \\ 4 & -3 & 0 & -2 \end{array}\right] .\]

.

Answer:

\[\operatorname{det}\ \left[\begin{array}{cccc} -3 & -4 & 2 & -3 \\ 4 & -3 & 2 & -2 \\ 1 & 3 & 0 & 0 \\ 4 & -3 & 0 & -2 \end{array}\right] = -110 \]


Example 30 πŸ”—

Show how to compute the determinant of the matrix

\[A= \left[\begin{array}{cccc} -5 & 2 & -1 & 1 \\ -4 & 1 & 0 & -4 \\ 3 & 2 & 0 & 4 \\ 0 & 5 & -3 & 2 \end{array}\right] .\]

.

Answer:

\[\operatorname{det}\ \left[\begin{array}{cccc} -5 & 2 & -1 & 1 \\ -4 & 1 & 0 & -4 \\ 3 & 2 & 0 & 4 \\ 0 & 5 & -3 & 2 \end{array}\right] = -187 \]


Example 31 πŸ”—

Show how to compute the determinant of the matrix

\[A= \left[\begin{array}{cccc} 0 & 2 & 0 & -5 \\ -2 & 2 & -2 & 4 \\ -4 & 2 & 0 & 5 \\ -1 & -3 & 1 & 4 \end{array}\right] .\]

.

Answer:

\[\operatorname{det}\ \left[\begin{array}{cccc} 0 & 2 & 0 & -5 \\ -2 & 2 & -2 & 4 \\ -4 & 2 & 0 & 5 \\ -1 & -3 & 1 & 4 \end{array}\right] = -64 \]


Example 32 πŸ”—

Show how to compute the determinant of the matrix

\[A= \left[\begin{array}{cccc} 3 & 4 & 0 & 4 \\ -3 & 3 & 1 & -1 \\ 4 & 0 & 0 & -3 \\ -6 & -4 & 2 & -4 \end{array}\right] .\]

.

Answer:

\[\operatorname{det}\ \left[\begin{array}{cccc} 3 & 4 & 0 & 4 \\ -3 & 3 & 1 & -1 \\ 4 & 0 & 0 & -3 \\ -6 & -4 & 2 & -4 \end{array}\right] = 218 \]


Example 33 πŸ”—

Show how to compute the determinant of the matrix

\[A= \left[\begin{array}{cccc} 4 & -6 & -2 & 4 \\ -3 & -3 & -5 & -4 \\ 6 & -3 & -6 & 0 \\ 0 & 2 & 1 & 0 \end{array}\right] .\]

.

Answer:

\[\operatorname{det}\ \left[\begin{array}{cccc} 4 & -6 & -2 & 4 \\ -3 & -3 & -5 & -4 \\ 6 & -3 & -6 & 0 \\ 0 & 2 & 1 & 0 \end{array}\right] = 84 \]


Example 34 πŸ”—

Show how to compute the determinant of the matrix

\[A= \left[\begin{array}{cccc} -2 & -1 & 0 & -1 \\ 4 & -1 & 2 & -2 \\ -1 & 0 & 3 & 0 \\ 2 & 6 & -1 & 2 \end{array}\right] .\]

.

Answer:

\[\operatorname{det}\ \left[\begin{array}{cccc} -2 & -1 & 0 & -1 \\ 4 & -1 & 2 & -2 \\ -1 & 0 & 3 & 0 \\ 2 & 6 & -1 & 2 \end{array}\right] = -111 \]


Example 35 πŸ”—

Show how to compute the determinant of the matrix

\[A= \left[\begin{array}{cccc} 6 & 1 & -2 & 1 \\ 0 & 3 & 1 & 0 \\ -1 & -6 & -6 & -2 \\ 5 & -6 & 1 & 2 \end{array}\right] .\]

.

Answer:

\[\operatorname{det}\ \left[\begin{array}{cccc} 6 & 1 & -2 & 1 \\ 0 & 3 & 1 & 0 \\ -1 & -6 & -6 & -2 \\ 5 & -6 & 1 & 2 \end{array}\right] = 71 \]


Example 36 πŸ”—

Show how to compute the determinant of the matrix

\[A= \left[\begin{array}{cccc} 5 & 4 & 6 & 5 \\ -6 & -6 & 4 & -6 \\ 3 & 2 & 0 & 6 \\ 0 & -2 & 0 & 1 \end{array}\right] .\]

.

Answer:

\[\operatorname{det}\ \left[\begin{array}{cccc} 5 & 4 & 6 & 5 \\ -6 & -6 & 4 & -6 \\ 3 & 2 & 0 & 6 \\ 0 & -2 & 0 & 1 \end{array}\right] = -292 \]


Example 37 πŸ”—

Show how to compute the determinant of the matrix

\[A= \left[\begin{array}{cccc} 2 & 6 & 2 & -2 \\ -1 & 5 & 0 & 1 \\ 1 & -1 & 6 & -5 \\ 0 & -2 & 1 & 0 \end{array}\right] .\]

.

Answer:

\[\operatorname{det}\ \left[\begin{array}{cccc} 2 & 6 & 2 & -2 \\ -1 & 5 & 0 & 1 \\ 1 & -1 & 6 & -5 \\ 0 & -2 & 1 & 0 \end{array}\right] = 80 \]


Example 38 πŸ”—

Show how to compute the determinant of the matrix

\[A= \left[\begin{array}{cccc} -3 & 4 & 0 & -6 \\ 4 & -3 & 0 & 1 \\ -1 & 3 & 3 & 6 \\ 3 & 2 & 1 & 2 \end{array}\right] .\]

.

Answer:

\[\operatorname{det}\ \left[\begin{array}{cccc} -3 & 4 & 0 & -6 \\ 4 & -3 & 0 & 1 \\ -1 & 3 & 3 & 6 \\ 3 & 2 & 1 & 2 \end{array}\right] = -203 \]


Example 39 πŸ”—

Show how to compute the determinant of the matrix

\[A= \left[\begin{array}{cccc} -4 & 0 & 0 & 4 \\ -4 & -2 & -1 & -6 \\ -3 & 1 & 2 & 4 \\ 6 & 0 & 5 & -4 \end{array}\right] .\]

.

Answer:

\[\operatorname{det}\ \left[\begin{array}{cccc} -4 & 0 & 0 & 4 \\ -4 & -2 & -1 & -6 \\ -3 & 1 & 2 & 4 \\ 6 & 0 & 5 & -4 \end{array}\right] = 184 \]


Example 40 πŸ”—

Show how to compute the determinant of the matrix

\[A= \left[\begin{array}{cccc} -3 & -5 & -1 & 3 \\ 3 & -3 & 4 & 3 \\ 6 & 0 & -3 & -2 \\ 0 & -2 & -1 & 0 \end{array}\right] .\]

.

Answer:

\[\operatorname{det}\ \left[\begin{array}{cccc} -3 & -5 & -1 & 3 \\ 3 & -3 & 4 & 3 \\ 6 & 0 & -3 & -2 \\ 0 & -2 & -1 & 0 \end{array}\right] = 168 \]


Example 41 πŸ”—

Show how to compute the determinant of the matrix

\[A= \left[\begin{array}{cccc} 6 & -3 & -2 & -3 \\ 6 & -5 & -1 & 4 \\ 3 & 4 & 0 & -5 \\ -2 & 4 & 0 & 6 \end{array}\right] .\]

.

Answer:

\[\operatorname{det}\ \left[\begin{array}{cccc} 6 & -3 & -2 & -3 \\ 6 & -5 & -1 & 4 \\ 3 & 4 & 0 & -5 \\ -2 & 4 & 0 & 6 \end{array}\right] = -540 \]


Example 42 πŸ”—

Show how to compute the determinant of the matrix

\[A= \left[\begin{array}{cccc} 3 & 0 & 4 & -3 \\ -2 & -6 & -2 & -6 \\ 1 & 0 & 3 & 0 \\ 5 & -6 & 1 & 1 \end{array}\right] .\]

.

Answer:

\[\operatorname{det}\ \left[\begin{array}{cccc} 3 & 0 & 4 & -3 \\ -2 & -6 & -2 & -6 \\ 1 & 0 & 3 & 0 \\ 5 & -6 & 1 & 1 \end{array}\right] = -534 \]


Example 43 πŸ”—

Show how to compute the determinant of the matrix

\[A= \left[\begin{array}{cccc} -1 & -3 & 1 & -5 \\ -3 & 6 & -4 & 6 \\ 5 & 3 & -4 & -4 \\ 0 & -1 & -3 & 0 \end{array}\right] .\]

.

Answer:

\[\operatorname{det}\ \left[\begin{array}{cccc} -1 & -3 & 1 & -5 \\ -3 & 6 & -4 & 6 \\ 5 & 3 & -4 & -4 \\ 0 & -1 & -3 & 0 \end{array}\right] = 731 \]


Example 44 πŸ”—

Show how to compute the determinant of the matrix

\[A= \left[\begin{array}{cccc} 5 & -1 & 0 & -4 \\ -3 & 2 & 1 & -6 \\ 0 & 5 & 0 & 2 \\ 6 & 5 & 2 & -3 \end{array}\right] .\]

.

Answer:

\[\operatorname{det}\ \left[\begin{array}{cccc} 5 & -1 & 0 & -4 \\ -3 & 2 & 1 & -6 \\ 0 & 5 & 0 & 2 \\ 6 & 5 & 2 & -3 \end{array}\right] = -431 \]


Example 45 πŸ”—

Show how to compute the determinant of the matrix

\[A= \left[\begin{array}{cccc} -3 & 1 & 4 & -4 \\ 3 & -5 & 3 & -5 \\ 0 & 0 & 2 & 1 \\ -4 & 3 & 0 & -3 \end{array}\right] .\]

.

Answer:

\[\operatorname{det}\ \left[\begin{array}{cccc} -3 & 1 & 4 & -4 \\ 3 & -5 & 3 & -5 \\ 0 & 0 & 2 & 1 \\ -4 & 3 & 0 & -3 \end{array}\right] = -5 \]


Example 46 πŸ”—

Show how to compute the determinant of the matrix

\[A= \left[\begin{array}{cccc} -2 & 3 & -3 & 1 \\ 6 & 4 & 4 & 0 \\ 4 & -1 & 0 & 0 \\ -4 & -4 & -3 & 3 \end{array}\right] .\]

.

Answer:

\[\operatorname{det}\ \left[\begin{array}{cccc} -2 & 3 & -3 & 1 \\ 6 & 4 & 4 & 0 \\ 4 & -1 & 0 & 0 \\ -4 & -4 & -3 & 3 \end{array}\right] = 332 \]


Example 47 πŸ”—

Show how to compute the determinant of the matrix

\[A= \left[\begin{array}{cccc} -5 & 0 & 3 & -2 \\ -5 & -6 & -6 & 4 \\ 1 & -2 & 0 & 0 \\ -6 & 1 & 6 & -2 \end{array}\right] .\]

.

Answer:

\[\operatorname{det}\ \left[\begin{array}{cccc} -5 & 0 & 3 & -2 \\ -5 & -6 & -6 & 4 \\ 1 & -2 & 0 & 0 \\ -6 & 1 & 6 & -2 \end{array}\right] = 216 \]


Example 48 πŸ”—

Show how to compute the determinant of the matrix

\[A= \left[\begin{array}{cccc} -5 & -4 & -2 & -3 \\ -5 & -1 & 0 & 1 \\ 0 & -1 & -3 & 0 \\ 2 & 2 & -6 & -3 \end{array}\right] .\]

.

Answer:

\[\operatorname{det}\ \left[\begin{array}{cccc} -5 & -4 & -2 & -3 \\ -5 & -1 & 0 & 1 \\ 0 & -1 & -3 & 0 \\ 2 & 2 & -6 & -3 \end{array}\right] = -307 \]


Example 49 πŸ”—

Show how to compute the determinant of the matrix

\[A= \left[\begin{array}{cccc} 6 & -6 & 1 & 0 \\ -4 & -2 & 2 & 0 \\ -6 & 4 & -1 & -1 \\ -3 & 4 & 1 & 0 \end{array}\right] .\]

.

Answer:

\[\operatorname{det}\ \left[\begin{array}{cccc} 6 & -6 & 1 & 0 \\ -4 & -2 & 2 & 0 \\ -6 & 4 & -1 & -1 \\ -3 & 4 & 1 & 0 \end{array}\right] = -70 \]


Example 50 πŸ”—

Show how to compute the determinant of the matrix

\[A= \left[\begin{array}{cccc} -6 & -2 & -1 & 3 \\ -2 & 0 & 0 & -4 \\ 1 & 1 & 3 & -5 \\ -4 & 0 & 1 & 3 \end{array}\right] .\]

.

Answer:

\[\operatorname{det}\ \left[\begin{array}{cccc} -6 & -2 & -1 & 3 \\ -2 & 0 & 0 & -4 \\ 1 & 1 & 3 & -5 \\ -4 & 0 & 1 & 3 \end{array}\right] = -108 \]