## Unit 01 Percentage, Ratio and Proportion (General Mathematics 9 PCTB)

Notes, solutions of unit 01, General Mathematics 9, for Punjab Curriculum & Textbook Board (PCTB), Lahore.

After studying this unit , the students will be able to:

- Define
- a matrix with real entries and relate its rectangular layout (formation) with real life,
- rows and columns of a matrix,
- the order of a matrix,
- equality of two matrices.

- Define and identify row matrix, column matrix, rectangular matrix, square matrix, zero/null matrix, diagonal matrix, scalar matrix, identity matrix, transpose of a matrix, symmetric and skew-symmetric matrices.
- Know whether the given matrices are suitable for addition/subtraction.
- Add and subtract matrices.
- Multiply a matrix by a real number.
- Verify commutative and associative laws under addition.
- Define additive identity of a matrix.
- Find additive inverse of a matrix.
- Know whether the given matrices are suitable for multiplication.
- Multiply two (or three) matrices.
- Verify associative law under multiplication.
- Verify distributive laws.
- Show with the help of an example that commutative law under multiplication does not hold in general (i.e., $AB \neq BA$).
- Define multiplicative identity of a matrix.
- Verify the result $(AB)^t = B^t A^t$.
- Define the determinant of a square matrix.
- Evaluate determinant of a matrix.
- Define singular and non-singular matrices.
- Define adjoint of a matrix.
- Find multiplicative inverse of a non-singular matrix A and verify that $AA^{-1} = I = A^{-1}A$ where I is the identity matrix.
- Use adjoint method to calculate inverse of a non-singular matrix.
- Verify the result $(AB)^{-1} = B^{-1}A^{-1}$
- Solve a system of two linear equations and related real life problems in two unknowns using
- Matrix inversion method,
- Cramer’ s rule.

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