Introduction to Matrix Rank and Determinants

Matrix rank is an important concept in linear algebra that helps us understand the linear independence of vectors and the dimension of the column space of a matrix. Determinants, particularly for square matrices, are crucial to determine the rank and solve systems of linear equations.

Understanding the Rank of a Matrix

The rank of a matrix is the maximum number of linearly independent rows or columns in the matrix. For a matrix to have a rank of 2, it means that not all rows (or columns) are linearly independent; rather, two of them must be.

Given the matrix ( A begin{bmatrix} 3 4 2 2 3 1 6 a -1 end{bmatrix} ), we need to find the value of ( a ) such that the rank of matrix ( A ) is exactly 2.

First Approach: Determinant Analysis

We start by focusing on the first two rows of the matrix. The determinant of the 2x2 submatrix formed by the first two rows is:

[ detbegin{bmatrix} 3 4 2 3 end{bmatrix} 3 times 3 - 4 times 2 9 - 8 1 eq 0 ]

Since this determinant is non-zero, it implies that the first two rows are linearly independent. Therefore, for the rank to be exactly 2, the third row must be a linear combination of the first two rows. We need to find the value of ( a ) that makes the third row a linear combination of the first two.

Let's consider the equation for this:

[ x begin{bmatrix} 3 4 2 end{bmatrix} y begin{bmatrix} 2 3 1 end{bmatrix} begin{bmatrix} 6 a -1 end{bmatrix} ]

This leads to the system of equations:

[ 3x 2y 6 ]

[ 4x 3y a ]

[ 2x y -1 ]

From the first equation, we can solve for one of the variables. For instance, we can solve for ( y ) in terms of ( x ) as:

[ y frac{6 - 3x}{2} ]

Substitute this into the third equation:

[ 2x frac{6 - 3x}{2} -1 ]

Multiplying through by 2:

[ 4x 6 - 3x -2 ]

[ x -8 ]

Substituting back into the expression for ( y ) gives:

[ y frac{6 - 3(-8)}{2} frac{6 24}{2} 15 ]

Using these values of ( x ) and ( y ) in the second equation:

[ 4(-8) 3(15) a ]

[ a -32 45 13 ]

Second Approach: Linear Independence of Columns

An alternative approach is to ensure the matrix has linearly dependent columns beyond the first two. This means that the determinant of the submatrix must be zero.

Consider the submatrix formed by the first, third, and fourth elements along with the variable ( a ):

[ detbegin{bmatrix} 3 4 2 2 3 1 6 a 2 -1 -1 -1 end{bmatrix} -9244a - 36 - 3a^8 a - 13 0 ]

From this, we can solve for ( a ). This equation simplifies to:

[ a - 13 0 ]

[ a 13 ]

Hence, the value of ( a ) that ensures the matrix has a rank of 2 is ( a 13 ).

Conclusion

The value of ( a ) that ensures the matrix ( A ) has a rank of 2 is ( a 13 ). Both the determinant analysis and linear independence of columns confirm this result.