Math Problem Statement

Rank of matrix 3,-1,2 -6,2,4 -3,1,2

Solution

To find the rank of the matrix:

3 & -1 & 2 \\ -6 & 2 & 4 \\ -3 & 1 & 2 \end{bmatrix}$$ We will perform row operations to bring it to row echelon form. Let's first check the determinant of the 3x3 matrix: $$\text{det}(A) = 3 \cdot \begin{vmatrix} 2 & 4 \\ 1 & 2 \end{vmatrix} - (-1) \cdot \begin{vmatrix} -6 & 4 \\ -3 & 2 \end{vmatrix} + 2 \cdot \begin{vmatrix} -6 & 2 \\ -3 & 1 \end{vmatrix}$$ I can calculate this step-by-step if you'd like, but from experience, the determinant is zero. This suggests that the matrix does not have full rank (i.e., it is not of rank 3). We now perform row reduction to see the rank more clearly. After row-reducing the matrix, we find that the rank of the matrix is **2**. Would you like to see the row reduction steps or explore this in more detail? ### Related questions: 1. How do you perform row reduction to find the rank of a matrix? 2. What are the properties of a matrix with rank 2? 3. How is the determinant of a matrix related to its rank? 4. How do you compute the determinant of a 3x3 matrix? 5. What are the conditions for a matrix to be invertible? ### Tip: When performing row reduction, always aim to simplify the matrix by getting zeros below and above the pivot elements.

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Math Problem Analysis

Mathematical Concepts

Matrix Algebra
Rank of a Matrix
Row Reduction
Determinants

Formulas

Rank of a matrix is the maximum number of linearly independent rows or columns.
Determinant of a 3x3 matrix: det(A) = a(ei − fh) − b(di − fg) + c(dh − eg)

Theorems

Row Echelon Form Theorem
Properties of Determinants

Suitable Grade Level

Undergraduate Mathematics or Advanced High School (Grades 11-12)