Diagonalisation: check 1
Difficulty: ★★☆Info
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.
Diagonalisation: review variant 1
Difficulty: ★★☆Info
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.
Diagonalisation: check 2
Difficulty: ★★☆Info
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.
Diagonalisation: review variant 2
Difficulty: ★★☆Info
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.
Diagonalisation: check 3
Difficulty: ★★☆Info
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.
Diagonalisation: review variant 3
Difficulty: ★★☆Info
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.
Diagonalisation: check 4
Difficulty: ★★☆Info
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.
Diagonalisation: review variant 4
Difficulty: ★★☆Info
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.