Chapter 2 Systems of Linear Equations and Matrices

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Chapter 2Systems of Linear Equations and Matrices

• Section 2.5
• Matrix Inverses

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What is a Matrix Inverse?

• The inverse of a matrix is comparable to the
  reciprocal of a real number.
• The product of a matrix and its identity matrix
  is always the matrix itself.
• In other words, multiplying a matrix by its
  identity matrix is like multiplying a number by
  1.

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Multiplicative Identity

• The real number 1 is the multiplicative identity
  for real numbers
• for any real number a, we have
• a 1 1 a a
• In this section, we define a multiplicative
  identity matrix I that has properties similar to
  those of the number 1.
• We use the definition of this matrix I to find
  the multiplicative inverse of any square matrix
  that has an inverse.

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Identity Matrix

• If I is to be the identity matrix, both of the
  products AI and IA must equal A.
• The identity matrix only exists for square
  matrices.

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Examples of Identity Matrices
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Determining if Matrices are Inverses of Each Other

• Recall that a number multiplied by its
  multiplicative inverse yields a product of 1.
• Similarly, the product of matrix A and its
  multiplicative inverse matrix A (read
  A-inverse) is I, the identity matrix.
• So, to prove that two matrices are inverses of
  each other, show that their product, regardless
  of the order theyre multiplied, is always the
  identity matrix.

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Example 1

• Prove or disprove that the matrices below are
  inverses of each other.
• a.)
• b.)
• c.)

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Finding the Inverse of a Matrix
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Row Operations on Matrices
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Example 2

• Find the inverse, if it exists, for each matrix.
• a.) b.)
• c.)

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Shortcut for Finding the Inverse of a 2 x 2 Matrix

• If a matrix is of the form
• then the inverse can be found by
• calculating
• Note ad bc ? 0.

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Example 3

• Find the inverse of the matrix below using the
  shortcut method.

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Solution to Example 3
To find the inverse of the matrix use the
formula and simplify.
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Solution to Example 3 (continued)