Unit 1: Matrix Algebra - Subjective Questions

Elementary row operations are operations that do not change the solution set of a system of linear equations. They are:\n\n- Interchange: Exchange two rows, .\n- Scaling: Multiply a row by a nonzero scalar, , where .\n- Row replacement: Add a multiple of one row to another row, .\n\nA matrix is in row echelon form when:\n\n- All nonzero rows occur above zero rows.\n- The first nonzero entry in each nonzero row is to the right of the first nonzero entry in the preceding row.\n- All entries below a leading entry are zero.\n\nThese operations are used systematically to obtain echelon form or reduced row echelon form. The number of nonzero rows in echelon form gives the rank of the matrix.

Given\n\n\nApply the row operations\n\n\nThen\n\n\nInterchanging the second and third rows gives\n\n\nThere are two nonzero rows in echelon form. Therefore,\n

The rank of a matrix is the maximum number of linearly independent rows or columns of the matrix. Row rank and column rank are always equal.\n\nTwo common methods are:\n\n- Echelon-form method: Reduce the matrix using elementary row operations. The rank is the number of nonzero rows in the resulting echelon form.\n- Minor method: The rank is the order of the largest nonzero minor of the matrix. A minor of order is the determinant of an submatrix.\n\nFor example, if a matrix reduces to\n\nthen it has two nonzero rows and hence rank .

Elementary row operations are:\n\n- Interchange of two rows.\n- Multiplication of a row by a nonzero scalar.\n- Addition of a multiple of one row to another row.\n\nElementary column operations are the corresponding operations performed on columns. Both types of operations are reversible.\n\nThe rank of a matrix remains unchanged under either elementary row operations or elementary column operations because these operations preserve the linear dependence relationships among the rows or columns. However, row operations are generally preferred for solving systems of equations because they directly transform the coefficient or augmented matrix.

Let be a nonsingular square matrix, so that . The cofactor of the element is denoted by and is defined as\n\nwhere is the minor obtained by deleting the th row and th column.\n\nThe matrix of cofactors is\n\nThe transpose of this matrix is called the adjoint or adjugate of :\n\n\nA fundamental identity is\n\n\nSince , divide by to obtain\n\nTherefore,\n\nThis formula is valid only when is square and nonsingular.

For\n\nwe have\n\n\nThe adjoint of a matrix is\n\nThus,\n\n\nHence,\n\n\nVerification:\n

To find by the Gauss-Jordan method, form the augmented matrix , where is the identity matrix of the same order as .\n\nUse elementary row operations to transform the left-hand block into . The same operations transform the right-hand block into :\n\n\nThe procedure is:\n\n- Write the augmented matrix .\n- Use row operations to create leading 1s in the left block.\n- Make all entries above and below each leading 1 equal to zero.\n- Read the right-hand block as the inverse.\n\nIf the left block cannot be reduced to , then is singular and its inverse does not exist.

For a square matrix of order , the following statements are equivalent:\n\n- has an inverse.\n- .\n- is nonsingular.\n- .\n- The rows and columns of are linearly independent.\n- The homogeneous system has only the trivial solution.\n\nIf , then is singular, its rows or columns are linearly dependent, and\n\nIn this case, does not exist.

Write the augmented matrix as\n\n\nApply\n\nThen\n\n\nNext, apply :\n\n\nThus, , so . From the second equation, , giving . From the first equation, , giving .\n\nTherefore,\n

Consider the system , where is the coefficient matrix and is the augmented matrix. Let the number of unknowns be .\n\nThe system is consistent if and only if\n\n\nThe cases are:\n\n- If , the system has a unique solution.\n- If , the system has infinitely many solutions.\n- If , the system is inconsistent and has no solution.\n\nThese results follow from the fact that a row such as , where , represents the contradiction .

A homogeneous system has the form\n\nwhere the constant vector is zero. It always has the trivial solution . It has nontrivial solutions if and only if\n\nwhere is the number of unknowns. For a square system, this is equivalent to .\n\nA nonhomogeneous system has the form\n\nIt may have no solution, one solution, or infinitely many solutions. It is consistent if and only if\n\nIf this common rank equals the number of unknowns, the solution is unique; otherwise, infinitely many solutions exist.

Let be a square matrix. A nonzero vector is called an eigenvector of corresponding to the eigen-value if\n\nRearranging gives\n\n\nSince , the homogeneous system must have a nontrivial solution. Therefore, the coefficient matrix must be singular:\n\nThis equation is called the characteristic equation of . Its roots are the eigen-values of . For each eigen-value , substitute it into to obtain the corresponding eigenvectors.

The characteristic equation is\n\nThus,\n\nwhich gives\n\nTherefore,\n\n\nFor :\n\nso . An eigenvector is\n\n\nFor :\n\nso , or . An eigenvector is\n\n\nHence, the eigen-pairs are\n

The algebraic multiplicity of an eigen-value is the number of times it occurs as a root of the characteristic polynomial .\n\nThe geometric multiplicity of is the dimension of its eigenspace, which is the null space of :\n\n\nFor every eigen-value,\n\n\nIf the algebraic and geometric multiplicities are equal for every eigen-value, the matrix has a complete set of linearly independent eigenvectors and is diagonalizable. If the geometric multiplicity is smaller for some eigen-value, the matrix is not diagonalizable.

Let be eigenvectors corresponding to distinct eigen-values . Thus,\n\nSuppose that\n\n\nApply to both sides:\n\nSubtract times the original relation to obtain\n\n\nRepeating this argument and using induction shows that\n\nbecause the eigen-values are distinct. Hence, the eigenvectors are linearly independent.

The Cayley-Hamilton theorem states that every square matrix satisfies its own characteristic equation.\n\nIf the characteristic polynomial of an matrix is\n\nthen replacing by gives\n\n\nThe theorem is significant because it can be used to:\n\n- Express higher powers of in terms of lower powers.\n- Calculate when the constant term is nonzero.\n- Simplify matrix functions and recurrence relations.\n- Verify relationships involving the eigen-values of a matrix.

The characteristic polynomial is\n\nTherefore,\n\n\nAccording to the Cayley-Hamilton theorem,\n\nNow,\n\nThus,\n\nHence, the theorem is verified.

From the characteristic equation obtained earlier,\n\nSince is nonsingular, multiply the equation by :\n\nTherefore,\n\nso\n\nSubstituting gives\n\n\nThus,\n

Given\n\nwe have\n\n\nMultiply by :\n\n\nMultiply by again:\n\n\nFinally,\n\nTherefore,\n\nThis illustrates how the Cayley-Hamilton relation reduces every higher power of to a linear combination of and .

Let be an matrix with eigen-values , counted according to algebraic multiplicity. Then:\n\n- The determinant is the product of the eigen-values:\n\n- The trace is the sum of the eigen-values:\n\n\nThese results follow from the characteristic polynomial. Consequently:\n\n- is singular if and only if at least one eigen-value is zero.\n- The sum of the diagonal elements equals the sum of the roots of the characteristic polynomial.\n- The product of the eigen-values equals the constant determinant term, up to the usual sign convention used in the characteristic polynomial.