Unit 1: Matrix methods and linear systems - Subjective Questions

A matrix is a rectangular arrangement of numbers or functions in rows and columns. A matrix having rows and columns is called an matrix and is written as .

Common types of matrices:

• Row matrix: Contains one row, for example .
• Column matrix: Contains one column, for example .
• Square matrix: Has the same number of rows and columns, for example .
• Diagonal matrix: A square matrix whose non-diagonal entries are zero, for example .
• Scalar matrix: A diagonal matrix with equal diagonal entries, such as .
• Identity matrix: A diagonal matrix with every diagonal entry equal to , for example .
• Symmetric matrix: Satisfies .
• Skew-symmetric matrix: Satisfies .
• Singular matrix: Has ; otherwise, it is non-singular.

The elementary row operations are:

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

The corresponding elementary column operations are:

• .
• , where .
• .

Two matrices are said to be row-equivalent if one can be obtained from the other by a finite sequence of elementary row operations. Such operations preserve the solution set of a linear system when applied to its augmented matrix. Both row and column operations preserve the rank of a matrix.

Elementary operations are used to transform a matrix into row-echelon form, reduced row-echelon form, or a normal form such as

where and are products of elementary matrices and .

Start with

Apply the elementary row operations

This gives

Interchange the second and third rows:

Therefore, a row-echelon form is

There are exactly two nonzero rows, so

The rank is also less than because the second row of the original matrix is twice the first row.

The rank of a matrix is the maximum number of linearly independent rows or columns of . The row rank and column rank of a matrix are always equal.

Using minors:

• The rank is if at least one minor of order is nonzero.
• Every minor of order greater than must be zero.
• Thus,

Using row-echelon form:

• Reduce using elementary row operations.
• Count the number of nonzero rows, or equivalently the number of pivot positions.
• That number is the rank of .

For example, if the echelon form is

then it has two pivot rows, and hence .

The row-reduction method is generally more convenient for large matrices.

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

Observe that

Therefore,

This is a nontrivial linear relation with

Hence, the vectors are linearly dependent.

Alternatively, placing the vectors as columns gives

Since the third column is the sum of the first two columns, , which confirms that the three vectors are linearly dependent.

Let be an matrix whose columns are . The homogeneous system

is equivalent to

The following statements are equivalent:

• The columns of are linearly independent.
• The equation has only the trivial solution .
• Every column of is a pivot column.
• .

If , then there is at least one free variable, so has nontrivial solutions and the columns are linearly dependent.

The columns of span the column space , whose dimension is

By the rank-nullity theorem,

Thus, rank measures the number of independent column directions, while nullity measures the number of independent solutions of .

The Rouché-Capelli theorem states that the linear system

is consistent if and only if

where is the augmented matrix.

For a system with unknowns:

• Unique solution:

Every unknown corresponds to a pivot variable.

• Infinitely many solutions:

At least one unknown is a free variable.

• No solution:

In row reduction, this case produces a contradictory row such as

which represents the impossible equation .

The augmented matrix is

Apply

Then

Interchange and :

Now apply :

Back-substitution gives

Therefore,

A homogeneous system has the form

It is always consistent because is always a solution. This solution is called the trivial solution. Nontrivial solutions exist if

where is the number of unknowns.

For the given system, the second equation is twice the first, so there is only one independent equation:

Thus, and there are free variables. Let

Then

Hence,

Therefore, the system has infinitely many solutions, and its solution space has dimension .

Form the augmented matrix :

Use the operations

which gives

Next apply

obtaining

Finally, apply :

Therefore,

Verification:

For a square matrix , the cofactor of is

where is the minor obtained by deleting row and column . The adjugate of is the transpose of its cofactor matrix:

A fundamental identity is

If , divide both sides by :

Therefore,

Conditions for existence:

• must be square.
• .
• Equivalently, for an matrix.
• Its rows and columns must be linearly independent.

If , the matrix is singular and has no inverse.

A nonzero vector is an eigenvector of if

where is the corresponding eigenvalue. Equivalently,

A nonzero solution exists when

For the given matrix,

Thus,

so

For :

Hence , and an eigenvector is

For :

Hence , and an eigenvector is

Any nonzero scalar multiple of these vectors is also an eigenvector.

Since is upper triangular, its eigenvalues are its diagonal entries:

For :

gives and , while is free. Therefore,

For :

gives and . Therefore,

For :

gives and . Therefore,

The matrix has three distinct eigenvalues and hence three linearly independent eigenvectors. Therefore, it is diagonalizable.

Let be an matrix with eigenvalues , counted with algebraic multiplicity.

Principal properties:

• The eigenvalues are the roots of .
• Their sum equals the trace:

• Their product equals the determinant:

• If is triangular, its eigenvalues are its diagonal entries.
• If is non-singular, is not an eigenvalue.
• If is an eigenvalue of non-singular , then is an eigenvalue of .
• If is an eigenvalue of , then is an eigenvalue of .
• For a polynomial , is an eigenvalue of .
• and have the same eigenvalues.
• Similar matrices have the same eigenvalues.
• Eigenvectors corresponding to distinct eigenvalues are linearly independent.

Let be a real symmetric matrix, so . Let and be eigenvectors corresponding to distinct eigenvalues and :

where .

Consider the scalar . Using ,

Since is symmetric,

Using ,

Therefore,

which implies

Since ,

Thus, and are orthogonal. Hence, eigenvectors of a real symmetric matrix corresponding to distinct eigenvalues are mutually orthogonal.

Two square matrices and are similar if there exists a non-singular matrix such that

Their characteristic polynomials satisfy

Since ,

Taking determinants,

Because ,

Therefore, and have the same characteristic polynomial and hence the same eigenvalues with the same algebraic multiplicities.

Also,

Using the cyclic property of trace,

Thus, similarity preserves the characteristic polynomial, eigenvalues, determinant, and trace.

Let be an eigenvalue of an matrix .

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

For every eigenvalue,

A matrix is diagonalizable if there exists a non-singular matrix such that

where is diagonal. This occurs if and only if has linearly independent eigenvectors.

Equivalently, is diagonalizable if, for every eigenvalue,

and the sum of the geometric multiplicities is .

In particular, an matrix with distinct eigenvalues is always diagonalizable.

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

Let be an matrix with characteristic polynomial

The adjugate identity gives

Write the adjugate as a matrix polynomial:

Multiplying and comparing coefficients of equal powers of produces a sequence of matrix equations. Multiplying these equations by suitable powers of and adding them causes the intermediate terms to cancel. The resulting equation is

Therefore,

Thus, if the characteristic equation is

then replacing scalar powers by corresponding matrix powers yields

The characteristic polynomial of is

Therefore,

By the Cayley-Hamilton theorem,

Since , the matrix is invertible. Multiply the matrix equation by :

Hence,

Substituting ,

Indeed,

The characteristic polynomial is

By the Cayley-Hamilton theorem,

so

Multiplying by for gives the recurrence

This has the same recurrence as the Fibonacci numbers, where , , and . Hence,

Since and ,

Therefore,

Thus, the Cayley-Hamilton theorem reduces every higher power of to a linear combination of and .