Основные операции с матрицами: умножение на число, сложение и умножение матриц (15.1-15.24)

$3\left[\begin{smallmatrix} 1 & 2 \\ 1 & 2\end{smallmatrix}\right] -\left[\begin{smallmatrix} 3 & 2 \\ 3 & 2\end{smallmatrix}\right] - 4\left[\begin{smallmatrix} 0 & 1 \\ 0 & 1\end{smallmatrix}\right]$

$2\left[\begin{smallmatrix} [l||l|l]2 \\ 2 \\ 1\end{smallmatrix}\right] - 3\left[\begin{smallmatrix} 0 \\ 5 \\ 6\end{smallmatrix}\right]$

$2\left[\begin{smallmatrix} 1 & 8 & 7 & - 15 \\ 1 & - 5 & - 6 & 11\end{smallmatrix}\right] -\left[\begin{smallmatrix} 5 & 24 & - 7 & - 1 \\ - 1 & - 2 & 7 & 3\end{smallmatrix}\right]$

$\mathbf{A}_{18} + \mathbf{A}_{40}$. $\mathbf{A}_{18} = \left[\begin{smallmatrix} 1 & 2 \\ 3 & 4 \end{smallmatrix}\right], \mathbf{A}_{40} = \left[\begin{smallmatrix} 4 & -2 \\ -3 & 1 \end{smallmatrix}\right]$.

$\mathbf{A}_{28} - \mathbf{A}_{12}$. $\mathbf{A}_{28} = \left[\begin{smallmatrix} 1 & 3 \\ 3 & 1 \end{smallmatrix}\right], \mathbf{A}_{12} = \left[\begin{smallmatrix} 1 & 1 \\ 1 & 1 \end{smallmatrix}\right]$.

$2\mathbf{A}_{573} - \mathbf{A}_{571}$. $\mathbf{A}_{573} = \left[\begin{smallmatrix} 0 & 0 & 1 & 1 & 1 \\ 0 & 1 & 0 & 0 & 1 \end{smallmatrix}\right], \mathbf{A}_{571} = \left[\begin{smallmatrix} 1 & 0 & 1 & 1 & 1 \\ 0 & 1 & 1 & 1 & 1 \end{smallmatrix}\right]$.

$\frac{1}{2}\left(\mathbf{c}_{51} + \mathbf{c}_{52}\right)$. $\mathbf{c}_{51} = \left[\begin{smallmatrix} 1 \\ 1 \\ -1 \end{smallmatrix}\right], \mathbf{c}_{52} = \left[\begin{smallmatrix} 1 \\ -1 \\ 1 \end{smallmatrix}\right]$.

$A + B = B + A$

$A + (B + C) = (A + B) + C$

$\alpha (\beta A) = (\alpha \beta ) A$

$\alpha (A + B) = \alpha A + \alpha B$

$(\alpha + \beta ) A = \alpha A + \beta A$.

Можно ли умножить строку длины $m$ на столбец высоты $n$?

Можно ли умножить столбец высоты $n$ на строку длины $m$?

$\left[\begin{smallmatrix} 2 & - 3 & 0\end{smallmatrix}\right]\left[\begin{smallmatrix} 4 \\ 3 \\ 1\end{smallmatrix}\right]$

$\left[\begin{smallmatrix} 4 \\ 3 \\ 1\end{smallmatrix}\right] \left[\begin{smallmatrix} 2 & -3 & 0 \end{smallmatrix}\right]$

$\left[\begin{smallmatrix} 1 & 1 \\ 1 & 1\end{smallmatrix}\right]\left[\begin{smallmatrix} 3 & 5 \\ 5 & 9\end{smallmatrix}\right]$

$\mathbf{A}_{1} \mathbf{A}_{12}$. $\mathbf{A}_{1} = \left[\begin{smallmatrix} 1 & 0 \end{smallmatrix}\right], \mathbf{A}_{12} = \left[\begin{smallmatrix} 1 & 1 \\ 1 & 1 \end{smallmatrix}\right]$.

$\mathbf{A}_{4} \mathbf{A}_{395}$. $\mathbf{A}_{4} = \left[\begin{smallmatrix} 0 & 1 & 0 & 0 \end{smallmatrix}\right], \mathbf{A}_{395} = \left[\begin{smallmatrix} 1 & 4 & 3 \\ 0 & 3 & 2 \\ 0 & 1 & 0 \\ 0 & 2 & 1 \end{smallmatrix}\right]$.

$\mathbf{A}_{442} \mathbf{c}_{168}$. $\mathbf{A}_{442} = \left[\begin{smallmatrix} 3 & 3 & -4 & -3 \\ 0 & 6 & 1 & 1 \\ 5 & 4 & 2 & 1 \\ 2 & 3 & 3 & 2 \end{smallmatrix}\right], \mathbf{c}_{168} = \left[\begin{smallmatrix} 0 \\ 1 \\ 0 \\ 0 \end{smallmatrix}\right]$.

$\mathbf{A}_{110} \mathbf{A}_{12}$. $\mathbf{A}_{110} = \left[\begin{smallmatrix} 3 & 1 \\ 2 & 1 \\ 1 & 0 \end{smallmatrix}\right], \mathbf{A}_{12} = \left[\begin{smallmatrix} 1 & 1 \\ 1 & 1 \end{smallmatrix}\right]$.

$\mathbf{A}_{3} \mathbf{A}_{205}$. $\mathbf{A}_{3} = \left[\begin{smallmatrix} 1 & 1 & 1 \end{smallmatrix}\right], \mathbf{A}_{205} = \left[\begin{smallmatrix} 1 & 2 & 3 \\ 2 & 3 & 4 \\ 3 & 4 & 5 \end{smallmatrix}\right]$.

$\mathbf{A}_{436} \mathbf{c}_{166}$. $\mathbf{A}_{436} = \left[\begin{smallmatrix} 1 & 1 & 1 & 1 \\ 1 & 0 & 1 & 1 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 1 & 1 \end{smallmatrix}\right], \mathbf{c}_{166} = \left[\begin{smallmatrix} 1 \\ 1 \\ 1 \\ 1 \end{smallmatrix}\right]$.

$\mathbf{A}_{601} \mathbf{A}_{602}$. $\mathbf{A}_{601} = \left[\begin{smallmatrix} \lambda_1 & 0 & 0 & \cdots & 0 & 0 \\ 0 & \lambda_2 & 0 & \cdots & 0 & 0 \\ 0 & 0 & \lambda_3 & \cdots & 0 & 0 \\ \vdots & \vdots & \vdots & \ddots & \vdots & \vdots \\ 0 & 0 & 0 & \cdots & \lambda_{n-1} & 0 \\ 0 & 0 & 0 & \cdots & 0 & \lambda_{n} \end{smallmatrix}\right], \mathbf{A}_{602} = \left[\begin{smallmatrix} 1/\lambda_1 & 0 & 0 & \cdots & 0 & 0 \\ 0 & 1/\lambda_2 & 0 & \cdots & 0 & 0 \\ 0 & 0 & 1/\lambda_3 & \cdots & 0 & 0 \\ \vdots & \vdots & \vdots & \ddots & \vdots & \vdots \\ 0 & 0 & 0 & \cdots & 1/\lambda_{n-1} & 0 \\ 0 & 0 & 0 & \cdots & 0 & 1/\lambda_{n} \end{smallmatrix}\right]$.

$\mathbf{A}_{601} \mathbf{A}_{605}$. $\mathbf{A}_{601} = \left[\begin{smallmatrix} \lambda_1 & 0 & 0 & \cdots & 0 & 0 \\ 0 & \lambda_2 & 0 & \cdots & 0 & 0 \\ 0 & 0 & \lambda_3 & \cdots & 0 & 0 \\ \vdots & \vdots & \vdots & \ddots & \vdots & \vdots \\ 0 & 0 & 0 & \cdots & \lambda_{n-1} & 0 \\ 0 & 0 & 0 & \cdots & 0 & \lambda_{n} \end{smallmatrix}\right], \mathbf{A}_{605} = \left[\begin{smallmatrix} 0 & 0 & \cdots & 0 & 0 & \lambda_1 \\ 0 & 0 & \cdots & 0 & \lambda_2 & 0 \\ 0 & 0 & \cdots & \lambda_3 & 0 & 0 \\ \vdots & \vdots & \iddots & \vdots & \vdots & \vdots \\ 0 & \lambda_{n-1} & \cdots & 0 & 0 & 0 \\ \lambda_n & 0 & \cdots & 0 & 0 & 0 \end{smallmatrix}\right]$.

$\mathbf{A}_{605} \mathbf{A}_{601}$. $\mathbf{A}_{605} = \left[\begin{smallmatrix} 0 & 0 & \cdots & 0 & 0 & \lambda_1 \\ 0 & 0 & \cdots & 0 & \lambda_2 & 0 \\ 0 & 0 & \cdots & \lambda_3 & 0 & 0 \\ \vdots & \vdots & \iddots & \vdots & \vdots & \vdots \\ 0 & \lambda_{n-1} & \cdots & 0 & 0 & 0 \\ \lambda_n & 0 & \cdots & 0 & 0 & 0 \end{smallmatrix}\right], \mathbf{A}_{601} = \left[\begin{smallmatrix} \lambda_1 & 0 & 0 & \cdots & 0 & 0 \\ 0 & \lambda_2 & 0 & \cdots & 0 & 0 \\ 0 & 0 & \lambda_3 & \cdots & 0 & 0 \\ \vdots & \vdots & \vdots & \ddots & \vdots & \vdots \\ 0 & 0 & 0 & \cdots & \lambda_{n-1} & 0 \\ 0 & 0 & 0 & \cdots & 0 & \lambda_{n} \end{smallmatrix}\right]$.

$\left(\mathbf{A}_{206}\right)^{2}$. $\mathbf{A}_{206} = \left[\begin{smallmatrix} -1 & 1 & 1 \\ -5 & 21 & 17 \\ 6 & -26 & -21 \end{smallmatrix}\right]$.

$\left(\mathbf{A}_{200}\right)^{2}$. $\mathbf{A}_{200} = \left[\begin{smallmatrix} 0 & 0 & 1 \\ 0 & 1 & 0 \\ 1 & 0 & 0 \end{smallmatrix}\right]$.

$\left(\mathbf{A}_{617}\right)^{2}$. $\mathbf{A}_{617} = \left[\begin{smallmatrix} 1 & 1 & 1 & \ldots & 1 & 1 \\ 0 & 1 & 1 & \ldots & 1 & 1 \\ \ldots & \ldots & \ldots & \ldots & \ldots & \ldots \\ 0 & 0 & 0 & \ldots & 1 & 1 \\ 0 & 0 & 0 & \ldots & 0 & 1 \end{smallmatrix}\right]$.

$\left(\mathbf{A}_{645}\right)^{2}$ при $\varepsilon = e^{2\pi i / n}$. $\mathbf{A}_{645} = \left[\begin{smallmatrix} 1 & 1 & 1 & \ldots & 1 \\ 1 & \varepsilon & \varepsilon^2 & \ldots & \varepsilon^{n-1} \\ 1 & \varepsilon^2 & \varepsilon^4 & \ldots & \varepsilon^{2(n-1)} \\ \ldots \ldots \ldots \ldots & \ldots \ldots & \ldots \ldots & \ldots \ldots \\ 1 & \varepsilon^{n-1} & \varepsilon^{2(n-1)} & \ldots & \varepsilon^{(n-1)^2} \end{smallmatrix}\right]$.

существовало произведение $AB$

существовало произведение $BA$

существовали произведения $AB$ и $BA$?

$\alpha (AB) = (\alpha A) B$

$(AB) C = A(BC)$

$A(B + C) = AB + AC$

$(A + B) C = AC + BC$

$A(B + C + D) = AB + AC + AD$.

$\left[\begin{smallmatrix} 1 & 2 \\ 3 & 4\end{smallmatrix}\right]\left[\begin{smallmatrix} 1 & 2\end{smallmatrix}\right]\left[\begin{smallmatrix} 2 \\ 1\end{smallmatrix}\right]$

$\left[\begin{smallmatrix} 2 \\ 4\end{smallmatrix}\right]\left[\begin{smallmatrix} 1 & 2\end{smallmatrix}\right]\left[\begin{smallmatrix} 2 \\ 1\end{smallmatrix}\right]$

$\left[\begin{smallmatrix} 1 \\ 2 \\ 2 \end{smallmatrix}\right]\left[\begin{smallmatrix} 2 & 1\end{smallmatrix}\right]\left[\begin{smallmatrix} 2 \\ 4 \end{smallmatrix}\right]$

$\mathbf{A}_{2} \mathbf{A}_{8} \mathbf{c}_{8} \mathbf{A}_{2}$. $\mathbf{A}_{2} = \left[\begin{smallmatrix} -12 & 13 \end{smallmatrix}\right], \mathbf{A}_{8} = \left[\begin{smallmatrix} 13574 & 13647 \\ 28423 & 28523 \end{smallmatrix}\right], \mathbf{c}_{8} = \left[\begin{smallmatrix} -1 \\ 1 \end{smallmatrix}\right], \mathbf{A}_{2} = \left[\begin{smallmatrix} -12 & 13 \end{smallmatrix}\right]$.

$\left[\begin{smallmatrix} 1 & 1 \\ 1 & 1\end{smallmatrix}\right]^{n}$

$\left[\begin{smallmatrix} 1 & 1 & 1 \\ 0 & 0 & 0 \\ 0 & 0 & 0\end{smallmatrix}\right]^{n}$

$\left[\begin{smallmatrix} 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0\end{smallmatrix}\right]^{3}$

$\left(\mathbf{A}_{5}\right)^{n}$. $\mathbf{A}_{5} = \left[\begin{smallmatrix} 1 & -1 \\ 1 & -1 \end{smallmatrix}\right]$.

$\left(\mathbf{A}_{13}\right)^{n}$. $\mathbf{A}_{13} = \left[\begin{smallmatrix} 1 & 1 \\ 0 & 1 \end{smallmatrix}\right]$.

$\left(\mathbf{A}_{14}\right)^{n}$. $\mathbf{A}_{14} = \left[\begin{smallmatrix} 1 & 0 \\ 1 & 0 \end{smallmatrix}\right]$.

$\left(\mathbf{A}_{77}\right)^{n}$. $\mathbf{A}_{77} = \left[\begin{smallmatrix} \cos \alpha & -\sin \alpha \\ \sin \alpha & \cos \alpha \end{smallmatrix}\right]$.

$\left(\mathbf{A}_{601}\right)^{n}$. $\mathbf{A}_{601} = \left[\begin{smallmatrix} \lambda_1 & 0 & 0 & \cdots & 0 & 0 \\ 0 & \lambda_2 & 0 & \cdots & 0 & 0 \\ 0 & 0 & \lambda_3 & \cdots & 0 & 0 \\ \vdots & \vdots & \vdots & \ddots & \vdots & \vdots \\ 0 & 0 & 0 & \cdots & \lambda_{n-1} & 0 \\ 0 & 0 & 0 & \cdots & 0 & \lambda_{n} \end{smallmatrix}\right]$.

$\left(\mathbf{A}_{614}\right)^{n}$. $\mathbf{A}_{614} = \left[\begin{smallmatrix} 0 & 1 & 0 & \ldots & 0 \\ 0 & 0 & 1 & \ldots & 0 \\ \ldots & \ldots & \ldots & \ldots & . \\ 0 & 0 & 0 & \ldots & 1 \\ 1 & 0 & 0 & \ldots & 0 \end{smallmatrix}\right]$.

$\left(\mathbf{A}_{613}\right)^{n}$. $\mathbf{A}_{613} = \left[\begin{smallmatrix} 0 & 1 & 0 & \ldots & 0 & 0 \\ 0 & 0 & 1 & \ldots & 0 & 0 \\ \ldots & \ldots & \ldots & \ldots & \ldots & \ldots \\ 0 & 0 & 0 & \ldots & 0 & 1 \\ 0 & 0 & 0 & \ldots & 0 & 0 \end{smallmatrix}\right]$.

$\left[\begin{smallmatrix} 7 & - 4 \\ - 5 & 3\end{smallmatrix}\right]$

$\left[\begin{smallmatrix} 0 & & \lambda_{1} \\ \lambda_{n} & & 0\end{smallmatrix}\right]$

$\left[\begin{smallmatrix} 1 & 2 & 3\end{smallmatrix}\right]$

$\left[\begin{smallmatrix} 1 \\ 2 \\ 3 \\ 4\end{smallmatrix}\right]$

$\mathbf{A}_{9}$. $\mathbf{A}_{9} = \left[\begin{smallmatrix} 9 & -5 \\ -5 & 3 \end{smallmatrix}\right]$.

$\mathbf{A}_{390}$. $\mathbf{A}_{390} = \left[\begin{smallmatrix} 2 & 1 & 1 \\ 3 & 0 & 1 \end{smallmatrix}\right]$.

$\mathbf{A}_{544}$. $\mathbf{A}_{544} = \left[\begin{smallmatrix} 1 & 2 & 3 & 0 & 0 \\ 2 & 3 & 4 & 0 & 0 \\ 1 & 1 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 2 \\ 0 & 0 & 0 & 3 & 4 \end{smallmatrix}\right]$.

$\mathbf{A}_{632}$. $\mathbf{A}_{632} = \left[\begin{smallmatrix} 0 & 1 & 1 & \ldots & 1 & 1 & 1 \\ -1 & 0 & 1 & \ldots & 1 & 1 & 1 \\ -1 & -1 & 0 & \ldots & 1 & 1 & 1 \\ \ldots & \ldots & \ldots & \ldots & \ldots & \ldots & \ldots \\ -1 & -1 & -1 & \ldots & -1 & -1 & 0 \end{smallmatrix}\right]$.

$(\alpha A)^{T} = \alpha A^{T}$

$(AB)^{T} = B^{T} A^{T}$

$(ABC)^{T} = C^{T} B^{T} A^{T}$; 1$)(A + B)^{T} = A^{T} + B^{T}$.

$A = \mathbf{A}_{12}, B = \mathbf{A}_{5}$. $\mathbf{A}_{12} = \left[\begin{smallmatrix} 1 & 1 \\ 1 & 1 \end{smallmatrix}\right], \mathbf{A}_{5} = \left[\begin{smallmatrix} 1 & -1 \\ 1 & -1 \end{smallmatrix}\right]$.

$A = \mathbf{A}_{20}, B = \mathbf{A}_{16}$. $\mathbf{A}_{20} = \left[\begin{smallmatrix} 0 & -1 \\ 1 & 0 \end{smallmatrix}\right], \mathbf{A}_{16} = \left[\begin{smallmatrix} 1 & -1 \\ 1 & 1 \end{smallmatrix}\right]$.

$\left[A, B\right] = - \left[B, A\right]$

$\left[A, A\right] = O$

$\left[A, E\right] = \left[E, A\right] = O$

$\left[A, (B + C)\right] = \left[A, B\right] + \left[A, C\right]$

$A = \mathbf{A}_{12}, B = \mathbf{A}_{5}$. $\mathbf{A}_{12} = \left[\begin{smallmatrix} 1 & 1 \\ 1 & 1 \end{smallmatrix}\right], \mathbf{A}_{5} = \left[\begin{smallmatrix} 1 & -1 \\ 1 & -1 \end{smallmatrix}\right]$.

$A = \mathbf{A}_{20}, B = \mathbf{A}_{16}$. $15. 20)$:. $\mathbf{A}_{20} = \left[\begin{smallmatrix} 0 & -1 \\ 1 & 0 \end{smallmatrix}\right], \mathbf{A}_{16} = \left[\begin{smallmatrix} 1 & -1 \\ 1 & 1 \end{smallmatrix}\right]$.

$\left\{ A, B\right\} = \left\{ B, A\right\}$

$\left\{ A, A\right\} = A^{2}$

$\left\{ A, E\right\} = A$

$\left\{ A, (B + C)\right\} = \left\{ A, B\right\} + \left\{ A, C\right\}$.

$f(t) = t^{2} - 2t + 1, A =\left[\begin{smallmatrix} 1 & 1 \\ 1 & 1\end{smallmatrix}\right]$

$f(t) = t^{2} - 2t + 1, A =\left[\begin{smallmatrix} 1 & 1 \\ 0 & 1\end{smallmatrix}\right]$

$f(t) = t^{2} - 3t + 2, A = \mathbf{A}_{11}$. $\mathbf{A}_{11} = \left[\begin{smallmatrix} 0 & -2 \\ 1 & 3 \end{smallmatrix}\right]$.

$f(t) = (t - \varepsilon )^{2}, A = \mathbf{A}_{78}$. $\mathbf{A}_{78} = \left[\begin{smallmatrix} \varepsilon & 1 \\ -1 & \varepsilon \end{smallmatrix}\right]$.

$f(t) = t^{2} + t + 1, A = \mathbf{A}_{206}$. $\mathbf{A}_{206} = \left[\begin{smallmatrix} -1 & 1 & 1 \\ -5 & 21 & 17 \\ 6 & -26 & -21 \end{smallmatrix}\right]$.

$f(t) = t^{2} - t, A = \mathbf{A}_{230}$. $\mathbf{A}_{230} = \left[\begin{smallmatrix} -1 & 4 & 3 \\ -2 & 5 & 3 \\ 2 & -4 & -2 \end{smallmatrix}\right]$.

$f(t) = t^{2} + 2t - 3, A = \mathbf{A}_{214}$. $\mathbf{A}_{214} = \left[\begin{smallmatrix} 13 & 16 & 16 \\ -5 & -7 & -6 \\ -6 & -8 & -7 \end{smallmatrix}\right]$.

$(A + B)^{2} = A^{2} + 2AB + B^{2}$

$(A + B)(A - B) = (A - B)(A + B)$

$A^{2} - B^{2} = (A + B)(A - B)$

$(A + E)^{3} = A^{3} + 3A^{2} + 3A + E$.