Unit 1: Matrix methods and linear systems - Subjective Questions

A matrix is a rectangular arrangement of elements in rows and columns. A matrix having rows and columns is written as , where is the element in the th row and th column.

• Order: The order of is .
• Equality: Matrices and are equal when they have the same order and for every .
• Transpose: The transpose is obtained by interchanging rows and columns, so .
• Row matrix: A matrix of order , such as .
• Column matrix: A matrix of order .
• Square matrix: A matrix having the same number of rows and columns.
• Diagonal matrix: A square matrix in which all off-diagonal entries are zero.
• Identity matrix: A diagonal matrix with every diagonal entry equal to .
• Symmetric matrix: A matrix satisfying .
• Skew-symmetric matrix: A matrix satisfying .

The three elementary row operations are:

• Interchange two rows: .
• Multiply a row by a nonzero scalar: , where .
• Add a multiple of one row to another: .

A matrix is in row-echelon form when:

• All zero rows occur below nonzero rows.
• The leading nonzero entry of each successive row lies to the right of the leading entry in the preceding row.
• All entries below each leading entry are zero.

Elementary row operations preserve row equivalence and the solution set of the corresponding linear system. Applying them successively to eliminate entries below pivots produces row-echelon form. If each pivot is changed to and all entries above it are also eliminated, the result is the reduced row-echelon form.

The rank of a matrix , denoted by or , is the maximum number of linearly independent rows or columns of .

To determine rank by elementary operations:

1. Reduce to row-echelon form using elementary row operations.
2. Identify the pivot positions or nonzero rows.
3. Count the number of nonzero rows.

That number is the rank of . Elementary row operations do not alter rank. For example, if

then there are two nonzero rows, and hence .

For an matrix,

The rank of a matrix can be defined using its minors. A minor of order is the determinant of an submatrix.

A matrix has rank if:

• At least one minor of order is nonzero.
• Every minor of order greater than is zero.

Thus,

Comparison of methods:

• The minor method examines determinants of square submatrices and is useful for theoretical proofs or small matrices.
• The row-reduction method transforms the matrix into echelon form and counts nonzero rows.
• Row reduction is usually more efficient for large matrices because calculating all possible minors is laborious.
• Both methods give the same rank because row rank, column rank, and determinantal rank are equal.

Vectors are linearly dependent if there exist scalars , not all zero, such that

They are linearly independent if the equation

has only the trivial solution .

A practical test is to form a matrix whose columns are the given vectors. The vectors are linearly independent exactly when the rank equals the number of vectors. For vectors in , they are independent exactly when the determinant of the corresponding square matrix is nonzero.

Any set containing the zero vector is dependent, and any set of more than vectors in is necessarily dependent.

Form the linear combination

The vectors satisfy

Therefore,

This relation has coefficients , , and , which are not all zero. Hence the vectors are linearly dependent.

Equivalently, the matrix

has its third column equal to the sum of its first two columns. Therefore , confirming dependence.

For the linear system , let be the coefficient matrix and the augmented matrix. By the Rouché–Capelli theorem, the system is consistent if and only if

If the common rank is and the number of unknowns is , then:

• If , the system has a unique solution.
• If , the system has infinitely many solutions, with free variables.
• If , the system is inconsistent and has no solution.

During row reduction, inconsistency appears as a row of the form

which represents the contradiction .

The augmented matrix is

Apply and :

Interchange and :

Apply :

Back-substitution gives , , so , and , so .

Therefore,

A homogeneous linear system has the form

It is always consistent because is a solution, called the trivial solution.

If has columns, then:

• The system has only the trivial solution when .
• It has nontrivial solutions when .
• The number of free variables is .

For a square matrix of order , a nontrivial solution exists exactly when

This is because makes singular, so its columns are linearly dependent and its null space contains a nonzero vector. The solution space of is called the null space, and its dimension is the nullity of .

To find the inverse of an nonsingular matrix by the Gauss–Jordan method:

1. Form the augmented matrix .
2. Apply elementary row operations to transform the left block into .
3. Apply every operation simultaneously to the right block.
4. When the left block becomes , the right block is :

The method succeeds only when is nonsingular, or equivalently when and . If a zero row appears on the left before it becomes the identity matrix, is singular and has no inverse.

The result can be verified using

For a square matrix , the adjugate satisfies

If , division by gives

For

its determinant is

The cofactor matrix is

so its transpose, the adjugate, is

Therefore, provided ,

If , the matrix is singular and its inverse does not exist.

Let be an matrix. A scalar is an eigenvalue of if there exists a nonzero vector such that

The nonzero vector is an eigenvector corresponding to . Rearranging gives

A nonzero solution exists only if is singular. Therefore,

This equation is the characteristic equation, and its roots are the eigenvalues. For each eigenvalue , solve

to obtain its eigenvectors. The set of all such vectors together with the zero vector is the eigenspace

The characteristic equation is

Thus,

so

Hence the eigenvalues are and .

For :

which gives . Therefore, the eigenvectors are nonzero multiples of

For :

which gives . Therefore, the eigenvectors are nonzero multiples of

Important properties of the eigenvalues of a square matrix include:

• The sum of the eigenvalues, counted with algebraic multiplicity, equals the trace:
  
• Their product equals the determinant:
  
• A matrix is singular if and only if is one of its eigenvalues.
• If is an eigenvalue of , then is an eigenvalue of for a positive integer .
• If is invertible and is an eigenvalue of , then is an eigenvalue of .
• The matrices and have the same eigenvalues.
• Similar matrices have the same characteristic polynomial and eigenvalues.
• If is triangular, its eigenvalues are its diagonal entries.
• The eigenvalues of a real symmetric matrix are real.

Let the characteristic polynomial of an matrix be written as

If the eigenvalues are , counted with algebraic multiplicity, then

Expanding the product, the coefficient of is

From the determinant expansion of , the same coefficient is . Hence,

The constant term of is , while the factored form gives . Therefore,

These formulas include repeated eigenvalues according to their algebraic multiplicities.

Let and be eigenvectors corresponding to distinct eigenvalues and . Suppose

Applying gives

Multiplying the original relation by gives

Subtracting yields

Since and , it follows that . The original relation then implies . Thus and are linearly independent. The same argument extends by induction to any collection of eigenvectors corresponding to distinct eigenvalues.

Significance: An matrix with distinct eigenvalues has linearly independent eigenvectors and is therefore diagonalizable. If contains these eigenvectors as columns, then

where is diagonal.

For an eigenvalue :

• Its algebraic multiplicity, denoted , is the number of times occurs as a root of the characteristic polynomial.
• Its geometric multiplicity, denoted , is the dimension of its eigenspace:
  

They satisfy

An matrix is diagonalizable if and only if it has linearly independent eigenvectors. Equivalently, for every distinct eigenvalue,

and the sum of the geometric multiplicities is .

For example, the identity matrix has one eigenvalue with algebraic and geometric multiplicities both equal to , so it is diagonalizable. A Jordan block of order has algebraic multiplicity but geometric multiplicity , so it is not diagonalizable.

Cayley–Hamilton theorem: Every square matrix satisfies its own characteristic equation.

Let the characteristic polynomial of an matrix be

The theorem states that

Proof: From the adjugate identity,

Write the adjugate as a matrix polynomial:

Expanding the product and equating coefficients of equal powers of produces matrix identities involving the , , and coefficients . Multiplying these identities by suitable powers of and adding them causes the terms involving the to cancel. The remaining identity is

Hence , proving the theorem.

The characteristic polynomial is

Therefore,

By the Cayley–Hamilton theorem,

Since , the matrix is invertible. Multiplying the equation by gives

Thus,

Now,

Hence,

The characteristic polynomial of is

By the Cayley–Hamilton theorem,

or

This relation reduces every higher power to a linear combination of and . We claim that

For , the formula gives . Assume it is true for . Then

Using ,

Thus the formula holds by induction. Substituting the matrix gives

Therefore,

for every positive integer .