Matrices
Math Form 5 · 7 lessons
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## Matrices Fundamentals ### Definition A matrix is a rectangular array of numbers or elements arranged in rows and columns. It is used in various mathematical computations, such as solving systems of equations and performing transformations. - A matrix is represented as: $$A = bmatrix a_{11} & a_{12} & & a_{1n} a_{21} & a_{22} & & a_{2n} & & & a_{m1} & a_{m2} & & a_{mn} bmatrix$$ where $a_{ij}$ denotes the element in the $i$th row and $j$th column. ### Key Concepts - **Order of a Matrix**: Defined by the number of rows ($m$) and columns ($n$), represented as $m n$. - **Square Matrix**: A matrix where the number of rows equals the number of columns ($m = n$). - **Diagonal Matrix**: A matrix where Elements $a_{ii}$ in a square matrix that lie on the principal diagonal are not zero. - **Zero Matrix**: A matrix where all elements are zero. ### Important Properties - **Matrix Addition and Subtraction**: $$bmatrix a & b c & d bmatrix bmatrix e & f g & h bmatrix = bmatrix a e & b f c g & d h bmatrix$$ Matrices must have the same order to be added or subtracted. - **Scalar Multiplication**: $$kbmatrix a & b c & d bmatrix = bmatrix ka & kb kc & kd bmatrix$$ Each matrix element is multiplied by a scalar $k$. - **Matrix Multiplication**: $$m.n{A} x n.p{B} = m.p{AB} bmatrix a & b c & d bmatrix bmatrix e & f g & h bmatrix = bmatrix ae+bg & af+bh ce+dg & cf+dh bmatrix$$ Pendaraban matriks hanya didefinisikan apabila bilangan lajur dalam $A$ sepadan dengan bilangan baris dalam $B$. ### Essential Formulas - **Transpose of a Matrix**: $$(A^)_{ij} = A_{ji}$$ - **Identity Matrix**: $$I_n = bmatrix 1 & 0 & & 0 0 & 1 & & 0 & & & 0 & 0 & & 1 bmatrix$$ - **Inverse of a Matrix (for $2 2$ Matrices)**: The inverse of a $2 2$ matrix is given by: $$A^{-1} = 1{ad-bc} bmatrix d & -b -c & a bmatrix$$ where $A = bmatrix a & b c & d bmatrix$. - **Determinant of a $2 2$ Matrix**: The determinant of $A = bmatrix a & b c & d bmatrix$ is: $$(A) = ad - bc$$ - **Simple Algebraic Solution in Matrices**: Solve for $x$ in the equation: $$bmatrix x & 2 3 & 4 bmatrix + bmatrix 1 & 0 2 & 1 bmatrix = bmatrix 5 & 2 5 & 5 bmatrix$$ Solution: $$bmatrix x & 2 3 & 4 bmatrix = bmatrix 5 & 2 5 & 5 bmatrix - bmatrix 1 & 0 2 & 1 bmatrix$$ $$bmatrix x & 2 3 & 4 bmatrix = bmatrix 4 & 2 3 & 4 bmatrix$$ Thus, $x = 4$. ### Core Examples - **Basic example**: Given $A = bmatrix 1 & 2 3 & 4 bmatrix$ and $B = bmatrix 5 & 6 7 & 8 bmatrix$, find $A + B$. $$A + B = bmatrix 1+5 & 2+6 3+7 & 4+8 bmatrix = bmatrix 6 & 8 10 & 12 bmatrix$$ - **Advanced application**: Multiply $A = bmatrix 1 & 2 3 & 4 bmatrix$ by $B = bmatrix 2 & 0 1 & 3 bmatrix$. $$AB = bmatrix (1 2 + 2 1) & (1 0 + 2 3) (3 2 + 4 1) & (3 0 + 4 3) bmatrix = bmatrix 4 & 6 10 & 12 bmatrix$$ ### Related Theorems/Rules - **Associative Property of Matrix Multiplication**: $$(AB)C = A(BC)$$ - **Distributive Property**: $$A(B + C) = AB + AC$$ ### Common Pitfalls - Attempting to add or subtract matrices of different orders. - Assuming matrix multiplication is commutative ($AB BA$). - Forgetting to check if the determinant is non-zero before finding an inverse matrix. ### Related Topics - **Determinants and Inverses of Matrices** - **Linear Equations and Systems of Equations** ### Quick Review Questions - What is the result of multiplying $A = bmatrix 1 & 3 2 & 4 bmatrix$ by $B = bmatrix 2 & 0 1 & 5 bmatrix$? - How do you find the transpose of the matrix $A = bmatrix 4 & 5 6 & 7 bmatrix$? - Find the inverse of $A = bmatrix 2 & 1 5 & 3 bmatrix$, if it exists. - Solve for $x$ in the equation: $$bmatrix x & 1 2 & 3 bmatrix + bmatrix 3 & 2 1 & 1 bmatrix = bmatrix 5 & 3 3 & 4 bmatrix.$$
Matrices
Solve for $ x $ in the equation below: $$ \begin{bmatrix} x & 2 \\ 3 & 4 \end{bmatrix} + \begin{bmatrix} 1 & 0 \\ 2 & 1 \end{bmatrix} = \begin{bmatrix} 5 & 2 \\ 5 & 5 \end{bmatrix} $$
Why A (4) is correct:
When you add the two matrices on the left, the top-left element becomes x + 1 = 5. Solving for x gives x = 4. ✓
Why the others are wrong:
- B (3): This would make x + 1 = 4, which doesn't match the 5 in the result matrix.
- C (5): This would make x + 1 = 6, which is too large.
- D (2): This would make x + 1 = 3, which doesn't equal 5.
Which matrix operation is not possible for the matrices below: $$ \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix} \text{ and } \begin{bmatrix} 5 & 6 & 7 \end{bmatrix} $$
• Multiplication (A) is correct – Matrix multiplication requires the first matrix's columns to equal the second matrix's rows. Here, the first matrix is 2×2 (2 columns) but the second is 1×3 (1 row), so they don't match; multiplication fails.
• Addition (B) is wrong – Both matrices would need the same dimensions. While these don't match, addition simply isn't defined, but the question asks what's "not possible."
• Subtraction (C) is wrong – Like addition, subtraction requires matching dimensions and isn't possible here either.
• None of these (D) is wrong – Since multiplication genuinely is impossible, "none of these" doesn't fit.
The key: multiplication has a specific column-row compatibility rule that fails here, making it the clearest "not possible" operation.
What is the determinant of the matrix below: \\ $$ \begin{bmatrix} 3 & 8 \\ 4 & 6 \end{bmatrix} $$
Correct answer: A. -14
For a 2×2 matrix, the determinant is calculated as: (top-left × bottom-right) − (top-right × bottom-left).
Here: (3 × 6) − (8 × 4) = 18 − 32 = -14 ✓
Why the others are wrong:
- B (-16): Likely made an arithmetic error (like calculating 3 × 6 = 18, then 18 − 34 instead of 18 − 32)
- C & D (positive values): These ignore the negative sign that results from subtracting the larger product from the smaller one
If $$ \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix} \times X = \begin{bmatrix} 3 & 6 \\ 9 & 12 \end{bmatrix} $$, find $ X $.
# Explanation
A. 3 is correct.
When you multiply a matrix by a scalar (a single number), you multiply every element in the matrix by that number. Here, multiplying the first matrix by 3 gives: (1×3=3, 2×3=6, 3×3=9, 4×3=12), which matches the result matrix exactly.
B. 2 is wrong — multiplying by 2 would give 2, 4, 6, 8, not 3, 6, 9, 12.
C. 4 is wrong — multiplying by 4 would give 4, 8, 12, 16, which doesn't match.
D. 5 is wrong — multiplying by 5 would give 5, 10, 15, 20, which also doesn't match.
If $$ \begin{bmatrix} 2 & 4 \\ 6 & 8 \end{bmatrix} $$ is divided by $ X $ to yield $$ \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix} $$, find $ X $.
Correct Answer: A (2)
When you divide a matrix by a number, you divide each element by that number. Dividing the matrix [2 & 4 \ 6 & 8] by 2 gives [1 & 2 \ 3 & 4]: (2÷2=1, 4÷2=2, 6÷2=3, 8÷2=4). ✓
Why others are wrong:
- B (4): Would give [0.5 & 1 \ 1.5 & 2]—too small
- C (3): Would give non-integer results like 2÷3 ≈ 0.67
- D (5): Would give [0.4 & 0.8 \ 1.2 & 1.6]—also too small
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