模擬テスト6-3
[2]
次の行列の型を答えよ。
(1)
$(3,1)$
(2)
$
\left(
\begin{array}{cc}
18 & 1\\
2 & -1 \\
7 & 3
\end{array}
\right)
$
(3)
$
\left(
\begin{array}{cc}
3 & 1 & 1 \\
2 & 1 & 2
\end{array}
\right)
$
(4)
$
\left(
\begin{array}{cc}
2 \\
4
\end{array}
\right)
$
[2]
次のうち計算できるものは計算し、計算できないものは計算不能と答えよ。
(1)
$(1\ \ 2)\left(
\begin{array}{cc}
-1 \\
2
\end{array}
\right)
$
(2)
$\left(
\begin{array}{cc}
-1\\
1\\
1
\end{array}
\right)
\left(
\begin{array}{cc}
3 & -1 & 2
\end{array}
\right)
$
(3)
$
\left(
\begin{array}{cc}
1\\
2\\
3
\end{array}
\right)
\left(
\begin{array}{cc}
-2 & -3
\end{array}
\right)
$
(4)
$
\left(
\begin{array}{cc}
1 \\
-3
\end{array}
\right)
\left(
\begin{array}{cc}
1 & 1 \\
2 & 0
\end{array}
\right)
$
(5)
$
\left(
\begin{array}{cc}
0 & 0 \\
0 & 0
\end{array}
\right)
\left(
\begin{array}{cc}
1 \\
2
\end{array}
\right)
$
(6)
$
\left(
\begin{array}{cc}
1 & -1 \\
1 & 6
\end{array}
\right)
\left(
\begin{array}{cc}
1 & -2 & 1 \\
0 & -3 & 0
\end{array}
\right)
$
(7)
$\left(
\begin{array}{cc}
2 & -2 & 0 \\
1 & 3 & 0\\
1 & 1 & 1
\end{array}
\right)
\left(
\begin{array}{cc}
2 & -3 & 1 \\
0 & 1 & -2\\
0 & 1 & 3
\end{array}
\right)
$
(8)
$(-2)\left(
\begin{array}{cc}
0\\
3\\
-1
\end{array}
\right)
$
(9)
$\left(
\begin{array}{cc}
0\\
3\\
-1
\end{array}
\right)
\left(
\begin{array}{cc}
1 & 6 & -7 \\
2 & 0 & 6
\end{array}
\right)
$
(10)
$\left(
\begin{array}{cc}
2 & 5 \\
1 & 8 \\
2 & -4
\end{array}
\right)
\left(
\begin{array}{cc}
-1 \\
0
\end{array}
\right)
$
(11)
$\left(
\begin{array}{cc}
2 & 5 \\
1 & 8 \\
\end{array}
\right)
+
\left(
\begin{array}{cc}
-1 \\
0
\end{array}
\right)
$
(12)
$\left(
\begin{array}{cc}
0 \\
2 \\
\end{array}
\right)
+
\left(
\begin{array}{cc}
-3 \\
1
\end{array}
\right)
$
(13)
$
\displaystyle \frac{1}{2}\left(
\begin{array}{cc}
2 \\
-1 \\
5
\end{array}
\right)
+
(-2)\left(
\begin{array}{cc}
0 \\
1 \\
-1
\end{array}
\right)
$
(14)
$
\left(
\begin{array}{cc}
2 & -3 & 1 \\
0 & 1 & -2\\
0 & 1 & 3
\end{array}
\right)\left(
\begin{array}{cc}
2 \\
-1 \\
5
\end{array}
\right)
+
(-2)\left(
\begin{array}{cc}
0 \\
1 \\
-1
\end{array}
\right)
$
(15)
$
\left(
\begin{array}{cc}
0 & 1 & -1 \\
1 & 0 & 1
\end{array}
\right)\left(
\begin{array}{cc}
0 \\
3 \\
4
\end{array}
\right)
+
(-2)\left(
\begin{array}{cc}
0 \\
1 \\
-1
\end{array}
\right)
$
[3]$\ $(1)
$
A =\left(
\begin{array}{cc}
0 & -2 \\
1 & -1
\end{array}
\right),
B =\left(
\begin{array}{cc}
1 & 2 \\
3 & 4
\end{array}
\right)
$の時、$2(A-B)+(A-10B)+2(2A+10B)-(5A+9B)$の値を求めよ。
(2)
$
A =\left(
\begin{array}{cc}
2 & 4 \\
-2 & -6
\end{array}
\right),
B =\left(
\begin{array}{cc}
a & -1 \\
1 & a
\end{array}
\right),
C =\left(
\begin{array}{cc}
2 & 6 \\
4 & 12
\end{array}
\right)
$について、逆行列を持てばそれを求めよ。ただし、$a$は実数とする。
(3)
$
\left(
\begin{array}{cc}
a & a+1 \\
a+1 & 2a
\end{array}
\right)
$ が逆行列を持たない時、$a$の値はいくつか？
(4)
$
A =\left(
\begin{array}{cc}
a & 1 \\
1 & a
\end{array}
\right)
$の逆行列$A^{-1}$が存在するかを実数$a$の値で分類して答えよ。
(5)
$
A =\left(
\begin{array}{cc}
1 & 2\\
0 & 1
\end{array}
\right)
$の時、$A^5-2A^4+A^3+A$を求めよ。
(6)
$
A =\left(
\begin{array}{cc}
1 & 2 \\
3 & 5
\end{array}
\right),
B =\left(
\begin{array}{cc}
3 & 1 \\
2 & 6
\end{array}
\right),
C =\left(
\begin{array}{cc}
0 & 2 \\
0 & 0
\end{array}
\right)
$について、の時、$XA=B$を満たす行列$X$,$AYB=C$を満たす行列$Y$をそれぞれ求めよ。
(7)
行列$A,B$が共に逆行列を持つ時、$(AB)^{-1}=B^{-1}A^{-1}$が成り立つことを示せ。
(8)
$
A =\left(
\begin{array}{cc}
2 & -6 \\
-1 & 3
\end{array}
\right),
B =\left(
\begin{array}{cc}
2 & 4 \\
-1 & -2
\end{array}
\right)
$について、$A^n,B^n$をそれぞれ求めよ。
(9)
$
A =\left(
\begin{array}{cc}
1 & 1 \\
1 & 3
\end{array}
\right)
$について、$AX=X+A$となるような行列$X$を求めよ。
(10)
$
A =\left(
\begin{array}{cc}
1 & 2 \\
-1 & 4
\end{array}
\right),
P =\left(
\begin{array}{cc}
2 & 1 \\
1 & 1
\end{array}
\right)
$かつ、$B=P^{-1}AP$とする。$B$,$B^n$,$A^n$を求めよ。
(11)
$
A =\left(
\begin{array}{cc}
4 & -1 \\
5 & -2
\end{array}
\right),
P =\left(
\begin{array}{cc}
1 & 1 \\
a & b
\end{array}
\right)
$について、$P$が逆行列を持ち、
$P^{-1}AP = \left(
\begin{array}{cc}
-1 & 0 \\
0 & 3
\end{array}
\right)$とする。この時、$a,b,A^n$を求めよ。
(12)
$
A =\left(
\begin{array}{cc}
-2 & -1 \\
3 & 1
\end{array}
\right),
B =\left(
\begin{array}{cc}
-1 & 2 \\
1 & -1
\end{array}
\right)
$の時、$(B^{-1}A^2B)^{-1}B^{-1}A,\ \ B^2(A^{-1}B)^{-1}$を求めよ。