Matrices: แบบฝึกหัด

Let \(M = \begin{pmatrix} 1 & k \\ 0 & 1 \end{pmatrix}\). If \(M^2 = \begin{pmatrix} 1 & 16 \\ 0 & 1 \end{pmatrix}\), find the value of the constant \(k\).

Let \(A = \begin{pmatrix} 3 & -2 \\ 1 & 0 \end{pmatrix}\) and \(I\) be the \(2 \times 2\) identity matrix. If \(B = A^2 - 3A + 4I\), find the matrix \(B^{-1}\).

Given a \( 2 \times 2 \) matrix \( A \) satisfies the matrix equation \( A^2 - 5A + 7I = 0 \), where \( I \) is the \( 2 \times 2 \) identity matrix and \( 0 \) is the \( 2 \times 2 \) zero matrix. Which of the following expressions represents \( A^{-1} \)?

Given matrices $$A = \begin{pmatrix} 1 & 2 \\ 0 & 3 \end{pmatrix}$$ and $$B = \begin{pmatrix} 4 & 1 \\ 2 & 0 \end{pmatrix}$$. Find the product $$AB$$.

Given matrix \( M = \begin{pmatrix} 3 & -1 \\ 2 & 1 \end{pmatrix} \). If \( M^2 + kM + 5I = 0 \), where \( I \) is the \( 2 \times 2 \) identity matrix and \( 0 \) is the \( 2 \times 2 \) zero matrix, find the value of \( k \).

Given the matrix \(A = \begin{pmatrix} 1 & 2 \\ 0 & -1 \end{pmatrix}\), find the matrix product \(A^2\).

Given matrices \(A = \begin{pmatrix} 1 & 2 \\ 0 & 3 \end{pmatrix}\) and \(B = \begin{pmatrix} -1 & 1 \\ 4 & 0 \end{pmatrix}\), calculate \(A^2 + B\).

Given \(A = \begin{pmatrix} 5 & 2 \\ 1 & 3 \end{pmatrix}\), find its transpose \(A^T\).

Given matrices \(A = \begin{pmatrix} 2 & 1 \\ -1 & 3 \end{pmatrix}\) and \(B = \begin{pmatrix} 0 & -2 \\ 4 & 5 \end{pmatrix}\).
(a) Find the matrix product \(AB\).
(b) Find the matrix \(C\) such that \(C = 2A + B^T\), where \(B^T\) is the transpose of \(B\).

Consider the matrix \( A = \begin{pmatrix} 2 & 1 \\ 0 & 2 \end{pmatrix} \) and the matrix \( B = \begin{pmatrix} 1 & 0 \\ 1 & 1 \end{pmatrix} \).
(a) Express \( A^2 \) in the form \( \lambda A + \mu I \), where \( \lambda \) and \( \mu \) are real constants and \( I \) is the \( 2 \times 2 \) identity matrix.
(b) Using the result of (a), find the inverse matrix \( A^{-1} \).
(c) Find the matrix \( (AB)^T \), where \( (AB)^T \) denotes the transpose of the product \( AB \).
(d) Evaluate \( \det(2(A^T B)^{-1}) \).