1.2 Law of Determinants 行列式的性质 - 独中课程 | 线上补习

高二数学 | 高级数学

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1.2 Law of Determinants 行列式的性质

[1]

Calculate the values of the following determinant.
计算下列各行列式之值

[a]

$\left|\begin{matrix}x^2-4&x^2-x-2\\x^2-1&x^2-2x+1\\\end{matrix}\right|$

[b]

$\left|\begin{matrix}a^2&a&1\\1&a^2&a\\a&1&a^2\\\end{matrix}\right|$

[c]

$\left|\begin{matrix}1&1&1\\a&b&c\\1-2a&1-2b&1-2c\\\end{matrix}\right|$

[d]

$\left|\begin{matrix}4&7&1&8\\0&2&4&6\\4&2&-1&5\\2&-5&-1&-3\\\end{matrix}\right|$

[e]

$\left|\begin{matrix}0&a&a&a\\a&0&a&a\\a&a&0&a\\a&a&a&0\\\end{matrix}\right|$

(a) –3(x $^2$ – 1)(x – 2)

(b) (a $^3$ – 1) $^2$

(d) 0

(e) –3a $^4$

[2]

Solve the following equation
解下列方程式

[a]

$\left|\begin{matrix}3x-4&-1&4\\2x-5&1&-2\\x+1&2&6\\\end{matrix}\right|=0$

[b]

$\left|\begin{matrix}x&x&x\\-2&x-2&0\\0&2&6-2x\\\end{matrix}\right|=0$

(a) 2

(b) 0, 1, 2

[3]

Prove the following identities :

[a]

$\left|\begin{matrix}x&y&z\\x^2&y^2&z^2\\x^3&y^3&z^3\\\end{matrix}\right|$ = $xyz(y-z)(z-x)(x-y)$

[b]

$\left|\begin{matrix}x+z&z&x\\y&x+y&x\\y&z&y+z\\\end{matrix}\right|$ = $4xyz$

[c]

$\left|\begin{matrix}1&a&bc\\1&b&ac\\1&c&ab\\\end{matrix}\right|$ = $(b-c)(c-a)(a-b)$

[d]

$\left|\begin{matrix}b+c&q+r&y+z\\c+a&r+p&z+x\\a+b&p+q&x+y\\\end{matrix}\right|$ = $2\left|\begin{matrix}a&b&c\\p&q&r\\x&y&z\\\end{matrix}\right|$