Matrices and the Solution of Linear Equations - Essay Example

For example in a square submatrix of order 33 of [B], namely, det(D) = 0Now, there is a square submatrix of the augmented matrix [B], namely, Such that det(E)  12.0  0. Hence, the rank of the augmented matrix [B] = 3.

Similarly, we have to find the rank of the coefficient matrix [A]. Det(A) = 0.Therefore, the coefficient matrix [A]’s rank is less than 3. A square submatrix of the coefficient matrix [A] is and its det (J) = -3  0. Hence, the rank of the coefficient matrix is 2. These systems of equations are inconsistent because the rank of the coefficient matrix [A] < rank of the augmented matrix [B]. Consequently, no solution exists for [A] [X] = [C]. (Sarachik, 1997 .P. 243)Part B It is required to find if this system of equations is consistent.

The coefficient matrix and the right-hand side of this matrix equation are given by The augmented matrix Does not have any 4th-order sub-matrices, hence the rank of [B] cannot exceed 3. One of the 3rd-order square submatrices of [B] is in which det(D) = 0. Moreover, a 3rd order square sub-matrices of [B], namely, and are such that their corresponding determinants are zero. In other words, all the 3rd-order square sub-matrices of the augmented matrix [B] have a zero determinant. Therefore, the rank of the augmented matrix [B] is less than 3.

Since, there exists a 2nd-order square sub-matrix of matrix [B], such that det(H) = -120≠0, the rank of the augmented matrix [B] is 2. The rank of the coefficient matrix [A], where is to be determined. Det(A) = 0. Hence, the rank of the coefficient matrix [A] is less than 3. is a square sub-matrix of matrix [A], and det (J) = -3  0. Therefore, the rank of the coefficient matrix [A] is 2. Hence, the rank of the coefficient matrix [A] = rank of the augmented matrix [B], accordingly the system of equations given by [A] [X] = [B] is consistent.

Nevertheless, the rank of the coefficient matrix [A] = rank of the augmented matrix [B] < 3, therefore there are only 2 constraints; consequently, these equations have infinitely many solutions.