Unit 6: Matrices and determinants - Subjective Questions

A matrix is a rectangular arrangement of numbers, symbols, or expressions arranged in rows and columns, enclosed within brackets. A matrix of order has rows and columns.

Types of matrices:

• Row matrix: Has only one row. Example:
• Column matrix: Has only one column. Example:
• Square matrix: Number of rows equals number of columns. Example:
• Diagonal matrix: A square matrix with all non-diagonal elements zero. Example:
• Scalar matrix: A diagonal matrix with equal diagonal elements. Example:
• Identity (unit) matrix: A scalar matrix with diagonal elements . Example:
• Zero (null) matrix: All elements are zero. Example:
• Upper/Lower triangular matrix: All elements below/above the main diagonal are zero.

Symmetric Matrix:

• A square matrix is symmetric if , i.e., for all .
• The elements are symmetric about the main diagonal.
• Example:

Skew-Symmetric Matrix:

• A square matrix is skew-symmetric if , i.e., .
• All diagonal elements must be zero since .
• Example:

Key difference: In symmetric matrices , while in skew-symmetric matrices with zero diagonal.

Rules for Addition and Subtraction:

• Two matrices can be added or subtracted only if they are of the same order.
• Corresponding elements are added or subtracted.
• If and , then .

Calculation of :

Calculation of :

Conditions for Matrix Multiplication:

• Two matrices and can be multiplied only if the number of columns of equals the number of rows of .
• If is of order and is of order , then is of order .
• Each element of the product is obtained by multiplying rows of with columns of and summing: .

Calculation:

Note: Matrix multiplication is generally not commutative, i.e., .

Transpose of a Matrix: The transpose of a matrix , denoted , is obtained by interchanging its rows and columns. If of order , then of order .

Properties of transpose:

Verification:

First,

So

Now, ,

Since , the property is verified.

Determinant of a matrix:
For , .

Determinant of a matrix (expansion along first row):
For ,

Calculation:

Hence .

Minor: The minor of an element is the determinant obtained by deleting the th row and th column of the matrix.

Cofactor: The cofactor of element is .

For :

Minors:

• (delete row 1, col 1)

Cofactors:

Singular Matrix: A square matrix is called singular if its determinant is zero, i.e., . A singular matrix does not have an inverse.

Non-singular Matrix: A square matrix is called non-singular if . Only non-singular matrices have an inverse.

For :

Since , matrix is singular.

For :

Since , matrix is non-singular.

Adjoint of a Matrix: The adjoint (or adjugate) of a square matrix is the transpose of the matrix of cofactors of .

Procedure:

• Find the cofactor of each element.
• Form the cofactor matrix .
• Take its transpose to get .

For a matrix , a shortcut is:

Calculation for :

Interchange diagonal elements and change sign of off-diagonal elements:

The inverse of a non-singular matrix is given by:

Step 1: Find

Since , the inverse exists.

Step 2: Find adj

Step 3: Compute

Step 1: Find (expand along first row):

Step 2: Find cofactors:

• , ,
• , ,
• , ,

Step 3: Form adj (transpose of cofactor matrix):

Step 4: Compute :

Write the system in matrix form :

Solution: .

Step 1: Find

Step 2: Find

Step 3: Compute

Hence and .

Matrix form :

Step 1:

Step 2: Cofactors and adj

Step 3:

Hence , , .

Cramer's Rule (two variables): For the system

let . If , then:

where and are obtained by replacing the respective column with the constants column.

Solution:

Therefore:

Determinant :

(replace col 1 by constants):

(replace col 2 by constants):

(replace col 3 by constants):

Solution:

Properties of Determinants:

• Row-Column interchange: The value of a determinant is unchanged if rows and columns are interchanged, i.e., .
• Interchange of two rows/columns: Interchanging any two rows (or columns) changes the sign of the determinant.
• Identical rows/columns: If any two rows (or columns) are identical, the determinant is zero.
• Scalar multiplication: If all elements of a row (or column) are multiplied by a constant , the determinant is multiplied by .
• Zero row/column: If any row (or column) has all zero elements, the determinant is zero.
• Sum property: If elements of a row are expressed as sums, the determinant can be split into a sum of two determinants.
• Invariance: Adding a multiple of one row (or column) to another does not change the value of the determinant.

Example: For because column 2 is twice column 1 (proportional columns give zero).

Step 1: Find

Step 2: Find adj

Step 3: Compute

Step 4: Compute

Since both sides are equal, the property is verified.

Matrix Inverse Method:

• Writes the system as and solves as .
• Requires computing the inverse .
• Suitable when the same coefficient matrix is used with multiple right-hand sides.
• Fails when (matrix is singular, inverse does not exist).

Cramer's Rule:

• Solves each variable directly as a ratio of determinants: .
• No need to compute the full inverse; only determinants are evaluated.
• Convenient for small systems (2 or 3 variables).
• Fails when .

Comparison summary:

Both methods require a non-singular coefficient matrix (unique solution exists only when the determinant is non-zero).

Compute :

Compute :

Conclusion: Since , matrix multiplication is not commutative.

Condition for existence: A square matrix has an inverse if and only if it is non-singular, i.e., .

Step-by-step procedure (adjoint method):

• Step 1: Compute the determinant . If , the inverse does not exist; stop.
• Step 2: Find the minor of each element by deleting its row and column.
• Step 3: Compute the cofactor of each element using .
• Step 4: Form the cofactor matrix .
• Step 5: Take the transpose of the cofactor matrix to obtain the adjoint: .
• Step 6: Compute the inverse using the formula:
  

Verification: The result can be checked by confirming , where is the identity matrix.