MATRICES

2x2 Matrix Questions (Singular, Inverse, Operations & Word Problems)

1. Matrix Operations (Addition and Subtraction)
   Given the matrices:
   \( A = \begin{bmatrix} 2 & -3 \\ 4 & 1 \end{bmatrix},\quad B = \begin{bmatrix} 5 & 0 \\ -2 & 3 \end{bmatrix} \)
   Find:
   a) \( A + B \)
   b) \( A - B \)
2. Matrix Multiplication
   Let \( A = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix},\quad B = \begin{bmatrix} 0 & 1 \\ -1 & 0 \end{bmatrix} \).
   Find the product \( AB \) and \( BA \). Are the two products equal?
3. Identity and Inverse
   Let \( A = \begin{bmatrix} 4 & 7 \\ 2 & 6 \end{bmatrix} \).
   a) Show that \( A \) is non-singular.
   b) Find the inverse of \( A \), denoted \( A^{-1} \).
   c) Verify that \( A \cdot A^{-1} = I \), the identity matrix.
4. Singular Matrix
   Let \( A = \begin{bmatrix} 2 & 4 \\ 1 & 2 \end{bmatrix} \).
   a) Find the determinant of \( A \).
   b) Explain why \( A \) is a singular matrix.
   c) What implication does this have when solving \( AX = B \)?
5. Word Problem: Book Sales
   A school sold 2 science books and 3 math books for TSh 19,000. On another day, they sold 4 science books and 2 math books for TSh 24,000.
   Let the price of a science book be \( x \) and a math book be \( y \).
   a) Form a matrix equation of the form \( AX = B \).
   b) Use the inverse matrix method to find \( x \) and \( y \).
6. Solving Simultaneous Equations using Inverse Matrix
   Solve the system:
   \( \begin{cases} 3x + 2y = 16 \\ 4x - y = 9 \end{cases} \)
   a) Express in matrix form \( AX = B \).
   b) Find \( A^{-1} \).
   c) Use it to solve for \( x \) and \( y \).
7. Matrix Equation
   Given:
   \( A = \begin{bmatrix} 2 & 1 \\ 5 & 3 \end{bmatrix},\quad X = \begin{bmatrix} x \\ y \end{bmatrix},\quad B = \begin{bmatrix} 7 \\ 24 \end{bmatrix} \)
   Solve for \( X \) using the inverse matrix method.
8. Word Problem: Farm Produce
   A farmer sells 3 bags of maize and 2 bags of rice for TSh 55,000. On another day, he sells 5 bags of maize and 4 bags of rice for TSh 95,000.
   Let the price per bag of maize be \( x \), and rice be \( y \).
   a) Form the matrix equation \( AX = B \).
   b) Use the inverse matrix method to find \( x \) and \( y \).
9. Determinants and Inverses
   Given \( A = \begin{bmatrix} a & b \\ c & d \end{bmatrix} \):
   a) Write the formula for \( \det(A) \).
   b) State the condition under which \( A \) is singular.
   c) Write the general formula for \( A^{-1} \), assuming \( A \) is non-singular.
10. Matrix Multiplication and Determinant
    Let \( A = \begin{bmatrix} 1 & 4 \\ 2 & 3 \end{bmatrix},\quad B = \begin{bmatrix} 0 & -1 \\ 1 & 2 \end{bmatrix} \).
    a) Find \( AB \) and \( \det(AB) \).
    b) Find \( \det(A) \) and \( \det(B) \).
    c) Verify that \( \det(AB) = \det(A) \cdot \det(B) \).
11. Word Problem: Ticket Sales
    At a concert, 2 adult tickets and 1 child ticket cost TSh 25,000.
    3 adult tickets and 2 child tickets cost TSh 43,000.
    Let the cost of an adult ticket be \( x \) and a child ticket be \( y \).
    Use the inverse matrix method to find \( x \) and \( y \).