9231 FP2 · Notes & worked examples
Diagonalisation · Short practice Check a final answer, then explain the reasoning on paper. Guest progress stays in this browser; an account saves future checks without overwriting your first result.
Recall
Eigenvalues and diagonalisation — objective 1 Maths4U Original · Short practice
\(\displaystyle A=\left[\begin{matrix}11 & 1 & -1\\- \frac{3}{2} & \frac{27}{2} & \frac{3}{2}\\- \frac{5}{2} & \frac{5}{2} & \frac{25}{2}\end{matrix}\right]\). A diagonalising matrix is \(Q=\left[\begin{matrix}1 & 1 & 0\\0 & 1 & 1\\1 & 0 & 1\end{matrix}\right]\), whose columns correspond to eigenvalues in increasing order.
Find the smallest eigenvalue.
Enter one number or fraction. Use at least six decimal places when needed; a distribution calculator may be used for probabilities.
Hint Worked solution
Fresh review variant
Eigenvalues and diagonalisation — objective 1 Maths4U Original · Short practice
\(\displaystyle A=\left[\begin{matrix}12 & 1 & -1\\- \frac{3}{2} & \frac{29}{2} & \frac{3}{2}\\- \frac{5}{2} & \frac{5}{2} & \frac{27}{2}\end{matrix}\right]\). A diagonalising matrix is \(Q=\left[\begin{matrix}1 & 1 & 0\\0 & 1 & 1\\1 & 0 & 1\end{matrix}\right]\), whose columns correspond to eigenvalues in increasing order.
Find the smallest eigenvalue.
Enter one number or fraction. Use at least six decimal places when needed; a distribution calculator may be used for probabilities.
Hint Worked solution
Method
Eigenvalues and diagonalisation — objective 2 Maths4U Original · Short practice
\(\displaystyle A=\left[\begin{matrix}12 & 1 & -1\\- \frac{3}{2} & \frac{29}{2} & \frac{3}{2}\\- \frac{5}{2} & \frac{5}{2} & \frac{27}{2}\end{matrix}\right]\). A diagonalising matrix is \(Q=\left[\begin{matrix}1 & 1 & 0\\0 & 1 & 1\\1 & 0 & 1\end{matrix}\right]\), whose columns correspond to eigenvalues in increasing order.
Find det A.
Enter one number or fraction. Use at least six decimal places when needed; a distribution calculator may be used for probabilities.
Hint Worked solution
Fresh review variant
Eigenvalues and diagonalisation — objective 2 Maths4U Original · Short practice
\(\displaystyle A=\left[\begin{matrix}13 & 1 & -1\\- \frac{3}{2} & \frac{31}{2} & \frac{3}{2}\\- \frac{5}{2} & \frac{5}{2} & \frac{29}{2}\end{matrix}\right]\). A diagonalising matrix is \(Q=\left[\begin{matrix}1 & 1 & 0\\0 & 1 & 1\\1 & 0 & 1\end{matrix}\right]\), whose columns correspond to eigenvalues in increasing order.
Find det A.
Enter one number or fraction. Use at least six decimal places when needed; a distribution calculator may be used for probabilities.
Hint Worked solution
Apply
Eigenvalues and diagonalisation — objective 3 Maths4U Original · Short practice
\(\displaystyle A=\left[\begin{matrix}13 & 1 & -1\\- \frac{3}{2} & \frac{31}{2} & \frac{3}{2}\\- \frac{5}{2} & \frac{5}{2} & \frac{29}{2}\end{matrix}\right]\). A diagonalising matrix is \(Q=\left[\begin{matrix}1 & 1 & 0\\0 & 1 & 1\\1 & 0 & 1\end{matrix}\right]\), whose columns correspond to eigenvalues in increasing order.
Find the (1,1) entry of A cubed.
Enter one number or fraction. Use at least six decimal places when needed; a distribution calculator may be used for probabilities.
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Fresh review variant
Eigenvalues and diagonalisation — objective 3 Maths4U Original · Short practice
\(\displaystyle A=\left[\begin{matrix}14 & 1 & -1\\- \frac{3}{2} & \frac{33}{2} & \frac{3}{2}\\- \frac{5}{2} & \frac{5}{2} & \frac{31}{2}\end{matrix}\right]\). A diagonalising matrix is \(Q=\left[\begin{matrix}1 & 1 & 0\\0 & 1 & 1\\1 & 0 & 1\end{matrix}\right]\), whose columns correspond to eigenvalues in increasing order.
Find the (1,1) entry of A cubed.
Enter one number or fraction. Use at least six decimal places when needed; a distribution calculator may be used for probabilities.
Hint Worked solution
Check understanding
Eigenvalues and diagonalisation — objective 4 Maths4U Original · Short practice
\(\displaystyle A=\left[\begin{matrix}14 & 1 & -1\\- \frac{3}{2} & \frac{33}{2} & \frac{3}{2}\\- \frac{5}{2} & \frac{5}{2} & \frac{31}{2}\end{matrix}\right]\). A diagonalising matrix is \(Q=\left[\begin{matrix}1 & 1 & 0\\0 & 1 & 1\\1 & 0 & 1\end{matrix}\right]\), whose columns correspond to eigenvalues in increasing order.
Find trace(A).
Enter one number or fraction. Use at least six decimal places when needed; a distribution calculator may be used for probabilities.
Hint Worked solution
Fresh review variant
Eigenvalues and diagonalisation — objective 4 Maths4U Original · Short practice
\(\displaystyle A=\left[\begin{matrix}15 & 1 & -1\\- \frac{3}{2} & \frac{35}{2} & \frac{3}{2}\\- \frac{5}{2} & \frac{5}{2} & \frac{33}{2}\end{matrix}\right]\). A diagonalising matrix is \(Q=\left[\begin{matrix}1 & 1 & 0\\0 & 1 & 1\\1 & 0 & 1\end{matrix}\right]\), whose columns correspond to eigenvalues in increasing order.
Find trace(A).
Enter one number or fraction. Use at least six decimal places when needed; a distribution calculator may be used for probabilities.
Hint Worked solution
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