Unit 3: Linear Algebra - Subjective Questions

Matrix: A matrix is a rectangular arrangement of numbers or elements in rows and columns. A matrix having rows and columns is said to have order .

For example,

is a matrix of order .

Common types of matrices:

• Row matrix: It has exactly one row, such as .
• Column matrix: It has exactly one column, such as .
• Square matrix: It has the same number of rows and columns.
• Diagonal matrix: It is a square matrix in which every non-diagonal entry is zero.
• Scalar matrix: It is a diagonal matrix whose diagonal entries are equal.
• Identity matrix: It is a scalar matrix whose diagonal entries are all , such as .
• Zero matrix: Every entry is zero.
• Symmetric matrix: It satisfies .
• Skew-symmetric matrix: It satisfies .

Matrix addition: Two matrices can be added only when they have the same order. If and , then

Scalar multiplication: If is a scalar, then

There is no restriction on the order of .

Matrix multiplication: If is of order and is of order , then is defined and has order . Its entries are

For example, if

then

Matrix multiplication is generally not commutative, so need not equal .

The transpose of a matrix is the matrix obtained by interchanging its rows and columns. Thus,

Let be an matrix and an matrix. The -th entry of is

The -th entry of is

Using and , we get

Therefore, the corresponding entries are equal, and hence

The rank of a matrix , denoted by , is the maximum number of linearly independent rows or columns of . Equivalently, it is the number of nonzero rows in a row-echelon form of .

To determine the rank:

• Apply elementary row operations to reduce the matrix to row-echelon form.
• The permitted operations are interchanging two rows, multiplying a row by a nonzero scalar, and adding a multiple of one row to another.
• Count the number of nonzero rows in the resulting echelon form.

For example,

Applying and gives

There are two nonzero rows. Therefore,

The determinant is a scalar associated with a square matrix. It indicates important properties such as whether the matrix is invertible.

Expanding along the first row,

Evaluating the determinants,

Thus,

Therefore,

Hence,

Important properties of determinants include:

• Transpose property: .
• Row interchange: Interchanging two rows changes the sign of the determinant.
• Equal or proportional rows: If two rows are equal or proportional, then the determinant is zero.
• Scalar multiplication: Multiplying one row by a scalar multiplies the determinant by .
• Row replacement: Adding a multiple of one row to another row does not change the determinant.
• Product property: For square matrices and , .
• Triangular matrix: The determinant of a triangular matrix is the product of its diagonal entries.
• Invertibility: A square matrix is invertible if and only if .

These properties simplify determinant calculations and help determine whether systems of equations have unique solutions.

Let be an matrix. The minor is the determinant obtained by deleting row and column . The corresponding cofactor is

The cofactor matrix is , and the adjugate of is

A fundamental identity is

If is nonsingular, then . Dividing the identity by gives

Therefore,

For a matrix

the formula becomes

provided .

A system of linear equations can be written in matrix form as

Homogeneous system:

• It has the form .
• It always has the trivial solution .
• It has nontrivial solutions if , where is the number of unknowns.
• If is a square matrix, nontrivial solutions exist when .

Nonhomogeneous system:

• It has the form , where .
• It is consistent if .
• It has a unique solution if this common rank equals the number of unknowns.
• It has infinitely many solutions if the common rank is less than the number of unknowns.
• It has no solution if .

Thus, a homogeneous system is always consistent, whereas a nonhomogeneous system may be consistent or inconsistent.

The augmented matrix is

Apply the row operations

This gives

Next apply :

Thus,

From the third row,

From the first row,

Therefore,

The Gauss-Jordan elimination method transforms the augmented matrix of a system into reduced row-echelon form using elementary row operations.

In reduced row-echelon form:

• Every nonzero row has a leading entry equal to .
• Each leading is the only nonzero entry in its column.
• Leading entries move to the right as the row number increases.
• Zero rows, if any, occur at the bottom.

If the coefficient matrix is transformed into the identity matrix, then

and the solution is read directly from .

Difference from Gaussian elimination:

• Gaussian elimination produces an upper triangular or row-echelon matrix and then uses back-substitution.
• Gauss-Jordan elimination continues eliminating entries both below and above every pivot.
• Gauss-Jordan elimination requires more arithmetic but gives the solution directly.
• It can also be used to calculate an inverse by reducing to .

The Rouché-Capelli theorem states that a system

is consistent if and only if

where is the coefficient matrix and is the augmented matrix.

Let

Then the system is classified as follows:

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

The theorem provides a complete rank-based criterion for deciding the consistency and number of solutions of a linear system.

Cramer's rule applies to a square system when . If is obtained by replacing the -th column of by , then

For the given system,

The determinant of the coefficient matrix is

Replacing the first column by ,

Therefore,

Replacing the second column by ,

Therefore,

Hence,

Let be an matrix. A nonzero vector is called an eigenvector of if

for some scalar . The scalar is called the corresponding eigenvalue.

Rearranging the equation gives

For this homogeneous system to have a nonzero solution, the matrix must be singular. Therefore,

This equation is called the characteristic equation.

The calculation proceeds as follows:

• Form the matrix .
• Evaluate .
• Solve the characteristic equation to obtain the eigenvalues.
• For each eigenvalue , solve .
• Every nonzero solution is an eigenvector corresponding to .

The characteristic equation is

Thus,

Expanding,



Therefore, the eigenvalues are

For ,

This gives . Hence the eigenvectors are the nonzero multiples of

For ,

This gives , so . Hence the eigenvectors are the nonzero multiples of

Therefore,

Let be eigenvectors of a matrix corresponding to distinct eigenvalues .

Suppose, for contradiction, that the eigenvectors are linearly dependent. Let be the first vector expressible as a linear combination of the preceding vectors:

Applying to both sides gives

Multiplying the original relation by gives

Subtracting these equations,

Since are linearly independent and the eigenvalues are distinct, every coefficient must be zero. This would imply , contradicting the definition of an eigenvector.

Therefore, eigenvectors corresponding to distinct eigenvalues are linearly independent.

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

The characteristic polynomial may be written as

Comparing this polynomial with

we obtain:

• Sum of eigenvalues:
  

• Product of eigenvalues:
  

Consequently, a matrix is singular if and only if at least one of its eigenvalues is zero. These relationships remain valid even when eigenvalues are repeated or complex.

A square matrix is diagonalizable if there exists an invertible matrix and a diagonal matrix such that

Equivalently,

The columns of are linearly independent eigenvectors of $A`, and the corresponding diagonal entries of $D$ are their eigenvalues. If

then

An matrix is diagonalizable if and only if it has linearly independent eigenvectors. In particular, a matrix with distinct eigenvalues is diagonalizable.

Diagonalization simplifies powers of matrices because

and powers of a diagonal matrix are easy to calculate.

LU decomposition expresses a square matrix as

where is lower triangular and is upper triangular.

In Doolittle's method, the diagonal entries of are . Let

Equating the entries of gives

These formulas require the relevant pivots and to be nonzero. Otherwise, row pivoting may be necessary.

Let , where has unit diagonal entries. Begin with

The first-column multipliers are

Eliminating the first-column entries gives


The next multiplier is

Eliminating the second-column entry gives

Therefore,

Verification by multiplication gives

Hence, the displayed matrices form the required LU decomposition.

Suppose the coefficient matrix has the decomposition

where is lower triangular and is upper triangular. Then the system

becomes

Introduce an intermediate vector such that

The original problem is then solved in two stages.

1. Forward substitution: Solve

Because is lower triangular, calculate from the first equation, then , and continue downward. In general,

2. Backward substitution: Solve

Because is upper triangular, begin with the last equation and proceed upward. In general,

LU decomposition is especially efficient when several systems have the same coefficient matrix but different right-hand sides, because the decomposition is calculated only once.