Compute the value of the constants, so that the equality is verified: $\begin{bmatrix} 1& 0 &a\\ 0& b &-2 \end{bmatrix}-3 \begin{bmatrix} 2& c &-4\\ 0&\frac{1}{3} &d \end{bmatrix}= \begin{bmatrix} e& 0 &8\\ 0& 2&7 \end{bmatrix}$

# Operation

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#### Matrix operations 2

Consider the identity matrix $I_3$ and the matrices $$B=\begin{bmatrix} 1&1&-1\\2& 0& 0\\-1&0&1 \end{bmatrix}\quad C=\begin{bmatrix} 1&0\\2& -1\\-1&1 \end{bmatrix}$$ Check which of the following operations are possible: $(a1)\ CB\quad (a2)\ B-CC^T\quad (a3)\ I_3+2BC$

# Operation

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#### Matrix equation 1

Let $A$, $B$, and $X$ be matrices with real values, and suppose that all the matrices involved have an inverse and the operations involved are all possible. Solve in order of $X$ the following matrix equation: $A (I-X^T)^T = (X^{-1} B)^{-1}$

# Equations

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#### Matrix equation 2

Let $A$, $B$, and $X$ be matrices with real values, and suppose that all the matrices involved have an inverse and the operations involved are all possible. Solve in order of $X$ the following matrix equation: $(A^{-1}X^{-1}B)^{-1}=(A^T+B)^T$

# Equations

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#### Matrix operations 3

Consider the matrices: $A=\begin{bmatrix} 1 & 2 & -1 & 3\\ 5 & 1 & 1 & 3 \end{bmatrix}$, $B=\begin{bmatrix} 0 & 1 & 2 & -1\\ 1 & 1 & 0 & 0 \end{bmatrix}$ and $C=\begin{bmatrix} 2 & 1\\ 1 & -1 \end{bmatrix}$. The matrix $M=\dfrac{1}{3} \left( BA^T\right)^T - C$ is:

# Operation

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#### Inverse of a matrix

It is known that a sequence of elementary operations on rows transforms the matrix $[A|I_3]$ into $\begin{bmatrix}1&3&0&|&1&2&3\\0&1&2&|&1&0&2\\0&0&1&|&2&3&1\\\end{bmatrix}$. The $A^{-1}$ matrix is:

# Inverse

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