End-of-chapter review exercise 20 · Short practice

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Recall

Eigenvalues and diagonalisation — objective 1

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 smallest eigenvalue.

Enter one number or fraction. Use at least six decimal places when needed; a distribution calculator may be used for probabilities.

Fresh review variant

Eigenvalues and diagonalisation — objective 1

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 smallest eigenvalue.

Enter one number or fraction. Use at least six decimal places when needed; a distribution calculator may be used for probabilities.

Method

Eigenvalues and diagonalisation — objective 2

Maths4U Original · Short practice

Find det A.

Enter one number or fraction. Use at least six decimal places when needed; a distribution calculator may be used for probabilities.

Fresh review variant

Eigenvalues and diagonalisation — objective 2

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 det A.

Enter one number or fraction. Use at least six decimal places when needed; a distribution calculator may be used for probabilities.

Apply

The characteristic equation — objective 3

Maths4U Original · Short practice

\(A=\left[\begin{matrix}15 & 1\\1 & 16\end{matrix}\right]\). Write the characteristic equation as \(\lambda^2-p\lambda+q=0\).

Find the (1,2) entry of A inverse.

Enter one number or fraction. Use at least six decimal places when needed; a distribution calculator may be used for probabilities.

Fresh review variant

The characteristic equation — objective 3

Maths4U Original · Short practice

\(A=\left[\begin{matrix}16 & 1\\1 & 17\end{matrix}\right]\). Write the characteristic equation as \(\lambda^2-p\lambda+q=0\).

Find the (1,2) entry of A inverse.

Enter one number or fraction. Use at least six decimal places when needed; a distribution calculator may be used for probabilities.

Check understanding

Parameter-dependent consistency — objective 4

Maths4U Original · Short practice

Let \(A(t)=\left[\begin{matrix}1 & 1 & 1\\1 & t - 14 & 2\\2 & 2 & t - 13\end{matrix}\right]\). In the singular case consider right-hand side \((17,33,c)^T\). For the separate nonsingular calculation set \(t=17\) and use \(b=\left[\begin{matrix}49\\99\\134\end{matrix}\right]\).

Find the determinant at \(\displaystyle t=32\).

Enter one number or fraction. Use at least six decimal places when needed; a distribution calculator may be used for probabilities.

Fresh review variant

Parameter-dependent consistency — objective 4

Maths4U Original · Short practice

Let \(A(t)=\left[\begin{matrix}1 & 1 & 1\\1 & t - 15 & 2\\2 & 2 & t - 14\end{matrix}\right]\). In the singular case consider right-hand side \((18,35,c)^T\). For the separate nonsingular calculation set \(t=18\) and use \(b=\left[\begin{matrix}52\\105\\142\end{matrix}\right]\).

Find the determinant at \(\displaystyle t=34\).

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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