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Matrix Inverses: Properties and Computation, Exams of Linear Algebra

University of Baltimore (UB)Linear Algebra

An in-depth exploration of matrix inverses, their computation using cofactor matrices, and essential properties. Topics include the inverse of a square matrix, unit matrix, diagonal matrix, orthogonal matrix, and upper/lower triangular matrices. Learn how to calculate the inverse of a 3 x 3 matrix and understand the significance of determinants and adjugate matrices.

What you will learn

  • What are the properties of the inverse of a matrix?
  • What is the inverse of a square matrix and how is it defined?
  • How to calculate the inverse of a 3 x 3 matrix using cofactor matrix?

Typology: Exams

2021/2022

Uploaded on 09/27/2022

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Lecture 5: Matrix Operations: Inverse
•Inverse of a matrix
•Computation of inverse using co-factor matrix
•Properties of the inverse of a matrix
•Inverse of special matrices
•Unit Matrix
•Diagonal Matrix
•Orthogonal Matrix
•Lower/Upper Triangular Matrices
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Download Matrix Inverses: Properties and Computation and more Exams Linear Algebra in PDF only on Docsity!

Lecture 5: Matrix Operations: Inverse

• Inverse of a matrix

  • Computation of inverse using co-factor matrix
  • Properties of the inverse of a matrix
  • Inverse of special matrices
    • Unit Matrix
    • Diagonal Matrix
    • Orthogonal Matrix
    • Lower/Upper Triangular Matrices

Matrix Inverse

• Inverse of a matrix can only be defined for square matrices.

• Inverse of a square matrix exists only if the determinant of that matrix

is non-zero.

• Inverse matrix of 𝐴 is noted as 𝐴

• Example:

  • 𝐴 = 2 − 1 1 0 , 𝐴

= 0 1 − 1 2 ,

  • 𝐴𝐴

= 2 − 1 1 0 0 1 − 1 2 = 𝐴

𝐴 = 0 1 − 1 2 2 − 1 1 0 = 1 0 0 1

Inverse of a 3 x 3 matrix (using cofactor matrix)

4

Matrix of Minors =

Cofactor of A (𝐂) =

adj A = CT =

A

  • 1

     1

|𝐴| ∗ adj A = 1 10

Properties of Inverse of a Matrix

• (A

  • 1
  • 1

= A

• (AB)

  • 1

= B

  • 1

A

  • 1

• (kA)

  • 1

= k

  • 1

A

  • 1

where k is a non-zero scalar.

• (A

T

  • 1

= (A

  • 1

T

Inverse of Diagonal matrices

  • The determinant of a diagonal matrix is the product of its diagonal elements.
  • If they all are non-zero, then determinant is non-zero and the matrix is invertible.
  • The inverse of a diagonal matrix A is another diagonal matrix B whose diagonal elements are the reciprocals of the diagonal elements of A.
  • Example:
    • 𝐴 =

, 𝐴 = 2 × 1 × − 1 = − 2 (≠ 0)

  • 𝐴 − 1

1 2

Inverse of Orthonormal matrices

• Earlier, we saw that multiplication of an orthogonal (orthonormal)

matrix in its transpose results in identity matrix

• If 𝐴 is an orthonormal matrix, its inverse is equal to its transpose

• 𝐴 is an orthonormal 𝑛 × 𝑛 matrix

• Recall 𝐴𝐴

, where 𝐼

is a 𝑛 × 𝑛 identity matrix

• So, 𝐴

Inverse of Upper/Lower Triangular Matrices

Upper Triangular Matrix:

𝐴 = 2 1 0 − 1 ; 𝐴 − 1 0.^5 𝑋 0 − 1 ; 2 1 0 − 1

  1. 5 𝑋 0 − 1 1 0 0 1 𝑆𝑜𝑙𝑣𝑖𝑛𝑔 𝑓𝑜𝑟 𝑋 𝑤𝑒 𝑔𝑒𝑡 𝑋 = 0. 5

Lower Triangular Matrix:

𝐵 = 2 0 2 1 ; 𝐵 − 1 =

  1. 5 0 𝑋 1 ; 2 0 2 1
  2. 5 0 𝑋 1 = 1 0 0 1 𝑆𝑜𝑙𝑣𝑖𝑛𝑔 𝑓𝑜𝑟 𝑋 𝑤𝑒 𝑔𝑒𝑡 𝑋 = − 1