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, number of columns, and the rank of the given matrices. Let’s analyze and solve them step by step.
Part (i)
Matrix:
23 \\ 5 \end{bmatrix}$$ - **Number of Rows**: 2 - **Number of Columns**: 1 - **Rank**: Since both rows contain non-zero elements and the matrix is of size $$2 \times 1$$, the rank is $$\mathbf{1}$$. --- ### Part (ii) Matrix: $$\begin{bmatrix} 2 & 1 & 4 \\ -2 & 6 & 5 \end{bmatrix}$$ - **Number of Rows**: 2 - **Number of Columns**: 3 - **Rank**: Reduce the matrix to row-echelon form to find the rank: $$\begin{bmatrix} 2 & 1 & 4 \\ 0 & 7 & 13 \end{bmatrix}$$ Since both rows are linearly independent, the rank is $$\mathbf{2}$$. --- ### Part (iii) Matrix: $$\begin{bmatrix} 7 & 5 \\ -1 & 35 \\ 8 & 1 \end{bmatrix}$$ - **Number of Rows**: 3 - **Number of Columns**: 2 - **Rank**: Using Gaussian elimination: After simplification, only two rows remain linearly independent. Hence, the rank is $$\mathbf{2}$$. --- ### Summary Table | Part | Rows | Columns | Rank | |------|------|---------|------| | (i) | 2 | 1 | 1 | | (ii) | 2 | 3 | 2 | | (iii)| 3 | 2 | 2 | Would you like detailed steps for Gaussian elimination or row reduction? --- Here are 5 related questions for practice: 1. What is the difference between rank and dimension of a matrix? 2. How do you determine if rows or columns of a matrix are linearly independent? 3. Explain the concept of row-echelon form in finding matrix rank. 4. What is the maximum rank of an $$m \times n$$ matrix? 5. Can the rank of a matrix exceed its number of rows? **Tip**: The rank of a matrix equals the number of leading 1s in its row-echelon form!Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Matrix Theory
Matrix Rank
Row and Column Analysis
Formulas
Rank of a matrix based on row-echelon form
Theorems
Linear Independence of Rows and Columns
Rank-Nullity Theorem
Suitable Grade Level
Grades 10-12