To find the rank of the given matrix, we can use the method of row reduction (Gaussian elimination) to transform the matrix into its row echelon form. The rank is then the number of non-zero rows in the echelon form.
Matrix Rank Calculation
The given matrix is:
$ A = \begin{bmatrix} 1 & 1 & 1 \\ 1 & -1 & -1 \\ 3 & 1 & 1 \end{bmatrix} $Step 1: Apply Row Operations
We apply elementary row operations to simplify the matrix:
- Subtract Row 1 from Row 2 ($R_2 \leftarrow R_2 - R_1$).
- Subtract 3 times Row 1 from Row 3 ($R_3 \leftarrow R_3 - 3R_1$).
Performing these operations:
$ \begin{bmatrix} 1 & 1 & 1 \\ 1-1 & -1-1 & -1-1 \\ 3-3(1) & 1-3(1) & 1-3(1) \end{bmatrix} = \begin{bmatrix} 1 & 1 & 1 \\ 0 & -2 & -2 \\ 0 & -2 & -2 \end{bmatrix} $Step 2: Further Row Reduction
Now, subtract Row 2 from Row 3 ($R_3 \leftarrow R_3 - R_2$):
$ \begin{bmatrix} 1 & 1 & 1 \\ 0 & -2 & -2 \\ 0-0 & -2-(-2) & -2-(-2) \end{bmatrix} = \begin{bmatrix} 1 & 1 & 1 \\ 0 & -2 & -2 \\ 0 & 0 & 0 \end{bmatrix} $Step 3: Identify Non-Zero Rows
The matrix is now in row echelon form. The number of non-zero rows determines the rank.
- Row 1: [1 1 1] (Non-zero)
- Row 2: [0 -2 -2] (Non-zero)
- Row 3: [0 0 0] (Zero)
There are 2 non-zero rows.
Conclusion
Therefore, the rank of the matrix is 2.
Important Questions from Matrices
Let A be a skew-symmetric matrix of order 3.
What is the value of det(4A4) - det(3A3) + det(2A2) - det(A) + det(-I) where I is the identity matrix of order 3?
The system of linear equations
x + 2y + z = 4, 2x + 4y + 2z = 8 and 3x + 6y + 3z = 10 has
- Let AX = B be a system of 3 linear equations with 3-unknowns. Let X 1 and X 2 be its two distinct solutions. If the combination aX 1 + bX 2 is a solution of AX = B; where a, b are real numbers, then which one of the following is correct ?
An ordered pair $(\alpha, \beta)$ for which the system of linear equations
$\alpha x + (\beta+1)y + z = 2$
$2\alpha x + (\beta+2)y + z = 3$
$\alpha x + \beta y + 2z = 2$ has a unique solution, isLet A and B be two non zero square matrics and AB and BA both are defined. It means
