考研线性代数：行列式部分初级专项练习题（2024 年） - 第7页共13页

题目 07

已知三阶矩阵 $A = (α, γ_{1}, γ_{2})$ , $B = (β, γ_{1}, γ_{2})$ , 其中 $α, β, γ_{1}, γ_{2}$ 是三维列向量, 且 $| A | = 3$ , $| B | = 4$ , 则 $| 5 A - 2 B | = ?$

可以根据题意举个特例，例如，令：

$A = [\begin{array}{lll} 3 \\ 1 & 1 \end{array}]$

$B = [\begin{array}{lll} 4 \\ 1 \end{array}]$

则：

$| 5 A - 2 B | = | \begin{array}{lll} 7 \\ 3 \\ 3 \end{array} | = 63.$

或者直接算：

$5 A - 2 B =$

$(5 α, 5 γ_{1}, 5 γ_{2}) - (2 β, 2 γ_{1}, 2 γ_{2}) =$

$(5 α - 2 β, 3 γ_{1}, 3 γ_{2})$

于是：

$| 5 A - 2 B | = | 5 α - 2 β, 3 γ_{1}, 3 γ_{2} | =$

$9 | 5 α - 2 β, γ_{1}, γ_{2} | =$

$9 (5 | α, γ_{1}, γ_{2} | - 2 | β, γ_{1}, γ_{2} |) =$

$9 (5 | A | - 2 | B |) = 9 (15 - 8) = 63.$