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Problem 276

Show that the matrix \(\mathrm{A}\) is not diagonalizable where $$ \mathrm{A}=\mid \begin{array}{cc} -3 & 2 \\ \mid-2 & 1 \mid \end{array} $$

Expert verified

Matrix A is given as:
$$
\mathrm{A}=\begin{pmatrix}
-3 & 2 \\
-2 & 1
\end{pmatrix}
$$
We find the characteristic equation:
$$
\det(\mathrm{A} - λI) = λ^2 + 2λ + 1 = (λ + 1)^2
$$
Solving the characteristic equation, we find that there is only one eigenvalue, λ = -1.
By calculating the eigenspace for λ = -1:
$$
\begin{pmatrix}
-2 & 2 \\
-2 & 2
\end{pmatrix}
$$
We find that there is only one linearly independent eigenvector:
$$
\begin{pmatrix}
1 \\
1
\end{pmatrix}
$$
Since matrix A is a 2x2 matrix and there is only one linearly independent eigenvector instead of the necessary two, the matrix A is not diagonalizable.

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Chapter 11

Verify the following rules by giving examples: (a) If \(\mathrm{A}\) is an \(\mathrm{n} \times \mathrm{n}\) diagonal matrix and \(\mathrm{B}\) is an \(\mathrm{n} \times \mathrm{n}\) matrix, each row of \(\mathrm{AB}\) is then just the product of the diagonal entry of \(A\) times the corresponding row of \(B\). (b) If \(\mathrm{B}\) is a diagonal matrix, each column of \(\mathrm{AB}\) is just the product of the corresponding column of \(\mathrm{A}\) with the corresponding diagonal entry of \(\mathrm{B}\).

Chapter 11

Show that the matrix \(\mathrm{A}\) is diagonalizable where $$ \mathrm{A}=\begin{array}{cll} 10 & 0 & 0 \\ 0 & 1 & 0 \\ \mid 1 & 0 & 1 \mid \end{array} $$

Chapter 11

Find a matrix P that diagonalizes $$ \mathrm{A}=\mid \begin{array}{rr} 3 & -2 \\ -2 & 3 \\ 0 & 0 \end{array} $$

Chapter 11

Let \(\mathrm{T}\) be the linear operator on \(\mathrm{R}^{2}\) such that and only if Compute the matrix of \(\mathrm{T}\) with respect to the basis \(\\{(1,1),(1,-1)\\}\)

Chapter 11

Define a symmetric matrix. Is every symmetric matrix similar to a diagonal matrix?

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