Engineering Mathematics II: Matrices and Determinants

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• Matrices and Determinants*

Introduction to Matrices

• A matrix is a rectangular array of numbers, symbols, or characters
• Elements of a matrix are arranged in rows and columns
• Each element is identified by its row and column index

Types of Matrices

• Singleton Matrix: a matrix with only one element (1 × 1)
• Row Matrix: a matrix with only one row (1 × n)
• Column Matrix: a matrix with only one column (m × 1)
• Rectangular Matrix: a matrix with a different number of rows and columns (m × n)
• Square Matrix: a matrix with equal number of rows and columns (n × n)
• Diagonal Matrix: a square matrix with all non-zero elements on the main diagonal
• Unit or Identity Matrix: a diagonal matrix with all main diagonal elements equal to 1
• Null or Zero Matrix: a matrix with all elements equal to 0
• Triangular Matrix: a square matrix with all elements above or below the main diagonal equal to 0
• Upper Triangular Matrix: a triangular matrix with all elements below the main diagonal equal to 0
• Lower Triangular Matrix: a triangular matrix with all elements above the main diagonal equal to 0
• Scalar Matrix: a diagonal matrix with all main diagonal elements equal to the same scalar

Matrix Operations

• Addition of Matrices: only possible for matrices of the same order
  • Two matrices can be added element-wise to form a new matrix
  • Commutative and associative laws hold
• Subtraction of Matrices: similar to addition, but with subtraction instead of addition
• Scalar Multiplication of Matrices: multiplying a matrix by a scalar
  • Properties: distributive law, associative law, and scalar multiplication holds
• Matrix Multiplication: only possible for matrices where the number of columns of the first matrix equals the number of rows of the second matrix
  • Properties: associative law, distributive law, and scalar multiplication holds

Equality of Matrices

• Two matrices are equal if all corresponding elements are equal
• Some properties of equality: symmetric, transitive, and reflexive

Transpose of a Matrix

• Interchange rows and columns to form the transpose of a matrix
• Properties of transposed matrices:
  • (A+B)T = AT + BT
  • (AB)T = BT AT
  • (kA)T = kAT
  • (AT)T = A### Matrices - Operations
• (A + B)T = AT + BT
• (AB)T = BT AT

Symmetric Matrices

• A square matrix is symmetric if it is equal to its transpose: A = AT
• When the original matrix is square, transposition does not affect the elements of the main diagonal

Identity, Diagonal, and Scalar Matrices

• The identity matrix, I, a diagonal matrix D, and a scalar matrix, K, are equal to their transpose since the diagonal is unaffected

Inverse of a Matrix

• The inverse of a scalar k is the reciprocal or division of 1 by the scalar: k-1 = 1/k
• The inverse of a square matrix, A, if it exists, is the unique matrix A-1 where: A A-1 = A-1 A = I
• Properties of the inverse:
  • (AB) -1 = B-1 A-1
  • (A -1) T = A T -1
  • (A -1) -1 = A
  • (kA) -1 = A -1 / k

Matrix Properties

• A square matrix that has an inverse is called a nonsingular matrix
• A matrix that does not have an inverse is called a singular matrix
• Square matrices have inverses except when the determinant is zero
• When the determinant of a matrix is zero, the matrix is singular

Matrix Operations

• The addition or subtraction of any two matrices is possible if the order of the two matrices is the same
• Multiplication of any two matrices is possible only when the number of columns in the first matrix equals the number of rows in the second matrix
• If the order of a matrix is "m × n", then its transpose matrix will be "n × m", where a transpose matrix is formed by changing the rows of a matrix into columns and its columns into rows