Math Problem Statement
Nyatakan bilangan baris, bilangan lajur dan peringkat matriks bagi setiap matriks yang berikut.
Solution
The question asks for the number of rows, the number of columns, and the rank of the matrices given. Let’s analyze each matrix:
i) Matrix:
23 \\ 5 \end{bmatrix}$$ - **Number of Rows**: 2 - **Number of Columns**: 1 - **Rank**: The matrix has only one column with non-zero entries, so the rank is 1. --- ### ii) Matrix: $$\begin{bmatrix} 2 & 1 & 4 \\ -2 & 6 & 5 \end{bmatrix}$$ - **Number of Rows**: 2 - **Number of Columns**: 3 - **Rank**: - Row reduce the matrix to check for linear independence: $$\begin{bmatrix} 2 & 1 & 4 \\ -2 & 6 & 5 \end{bmatrix} \sim \begin{bmatrix} 1 & 0.5 & 2 \\ 0 & 7 & 9 \end{bmatrix}$$ Both rows are linearly independent. Thus, the rank is 2. --- ### iii) Matrix: $$\begin{bmatrix} 7 & 5 \\ -1 & 35 \\ 8 & 1 \end{bmatrix}$$ - **Number of Rows**: 3 - **Number of Columns**: 2 - **Rank**: - Check linear independence by row reducing: $$\begin{bmatrix} 7 & 5 \\ -1 & 35 \\ 8 & 1 \end{bmatrix} \sim \begin{bmatrix} 7 & 5 \\ 0 & 40 \\ 0 & -39 \end{bmatrix}$$ The three rows are linearly independent, so the rank is 2. --- ### Summary: 1. i) Rows: 2, Columns: 1, Rank: 1 2. ii) Rows: 2, Columns: 3, Rank: 2 3. iii) Rows: 3, Columns: 2, Rank: 2 Would you like detailed steps for any matrix rank calculation or further explanation? --- **Five follow-up questions:** 1. What does it mean for rows to be linearly independent in a matrix? 2. How do we determine if a matrix is full rank? 3. Can a matrix have a rank higher than its number of rows or columns? 4. What is the geometric interpretation of matrix rank? 5. How does row-reduction help in determining the rank of a matrix? **Tip**: Always double-check the row-reduced form of a matrix for zero rows or dependent rows when determining its rank!Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Matrix rows and columns
Matrix rank
Linear independence
Formulas
Row reduction to echelon form for rank calculation
Theorems
The rank of a matrix is the number of linearly independent rows or columns
Suitable Grade Level
Grades 10-12