Eigenvalues and Eigenvectors

Eigenvalues and eigenvectors are foundational principles in the study of linear algebra. They act as crucial tools to break down intricate problems, predominantly in the realm of linear equation systems. In essence, an eigenvector is a non-zero vector, which when subjected to a linear transformation, only alters by a scalar factor. The associated eigenvalue is this particular scalar.

Finding the Eigenvalues

If the column vector $\left[\begin{array}{l}-2 \\ -2\end{array}\right]$ is an eigenvector of the $2 \times 2$ matrix $A=\left[\begin{array}{ll}a & 1 \\ 1 & 2\end{array}\right]$ corresponding to the eigenvalue $\lambda$, then $a+\lambda=$ ? (A) 7 (B) 4 (C) 8 (D) 5 (E) 6
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Finding the Characteristic Equation

Given the matrix \( A = \begin{bmatrix} 1 & 3 \ 4 & 2 \end{bmatrix} \), find the characteristic equation and hence the eigenvalues.
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Finding the Eigenvectors/Eigenspace of a Matrix

If the column vector $\left[\begin{array}{c}5 \\ -5\end{array}\right]$ is an eigenvector of the $2 \times 2$ matrix $A=\left[\begin{array}{cc}4 & a \\ -1 & 1\end{array}\right] \cdots$ corresponding to the eigenvalue $\lambda$, then $a+\hat{\lambda}=?$
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Popular Problems

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