Math Problem Statement
Determine the number of rows, columns, and rank of matrices, as well as identify specific elements in a given matrix.
Solution
Let's solve the questions step-by-step.
Part (a): Matrix Details
For each matrix, we need to state:
- Bilangan baris (number of rows).
- Bilangan lajur (number of columns).
- Peringkat matriks (rank of the matrix).
i) $$\begin{bmatrix}
2 & 3 \ -5 \end{bmatrix}$$
- Rows: 2 rows.
- Columns: 1 column.
- Rank: Since all rows are non-zero, the rank is 2.
ii) $$\begin{bmatrix}
2 & 1 & 4 \ -2 & 6 & 5 \end{bmatrix}$$
- Rows: 2 rows.
- Columns: 3 columns.
- Rank: Perform row reduction or determine linear independence. After evaluation, the rank is 2.
iii) $$\begin{bmatrix}
7 & 5 \ -1 & 3 & 5 \ 8 & 1 \end{bmatrix}$$
- Rows: 3 rows.
- Columns: 3 columns.
- Rank: After evaluating the rows for independence, the rank is 3.
Part (b): Specific Elements in Matrix
Given the matrix:
0 & 5 & 1 \\ -3 & 1 & 9 \\ 1 & -2 & -7 \end{bmatrix}$$ We need to determine: 1. $$a_{11}$$: The element in the 1st row, 1st column → $$0$$. 2. $$a_{23}$$: The element in the 2nd row, 3rd column → $$9$$. 3. $$a_{32}$$: The element in the 3rd row, 2nd column → $$-2$$. --- ### Final Answers: #### (a) - i) Rows: 2, Columns: 1, Rank: 2 - ii) Rows: 2, Columns: 3, Rank: 2 - iii) Rows: 3, Columns: 3, Rank: 3 #### (b) - $$a_{11} = 0$$, $$a_{23} = 9$$, $$a_{32} = -2$$ --- Would you like detailed steps for determining the rank or row reduction for any of these matrices? ### Related Questions: 1. How is the rank of a matrix determined in general? 2. Can a rank be greater than the number of rows or columns of a matrix? 3. What is the significance of the rank of a matrix in linear algebra? 4. Can you explain how to perform row reduction on a matrix? 5. What happens if a matrix contains all zeros? What is its rank? ### Tip: Remember that the rank of a matrix represents the maximum number of linearly independent rows or columns. Always simplify the matrix using row-reduction techniques to find the rank efficiently.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Matrix
Matrix Dimensions
Matrix Rank
Linear Algebra
Formulas
Rank of a matrix is the maximum number of linearly independent rows or columns.
Theorems
Linear Independence in Matrices
Row Reduction Method
Suitable Grade Level
Grade 11-12 (Pre-University)