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2.4 The Inverse of a Matrix

2.4 The Inverse of a Matrix. Inverse of A Inverse of a 2x2 Matrix Matrix With No Inverse Solving a Matrix Equation. Inverse of A. The inverse of a square matrix A , denoted by A -1 , is a square matrix with the property A -1 A = AA -1 = I ,

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2.4 The Inverse of a Matrix

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  1. 2.4 The Inverse of a Matrix • Inverse of A • Inverse of a 2x2 Matrix • Matrix With No Inverse • Solving a Matrix Equation

  2. Inverse of A • The inverse of a square matrix A, denoted by A-1, is a square matrix with the property • A-1A = AA-1 = I, • where I is an identity matrix of the same size.

  3. checks checks Example Inverse of A • Verify that is the inverse of

  4. Inverse of a 2x2 • To determine the inverse of if = ad - bc ≠ 0. • Interchange a and d to get • Change the signs of b and c to get • Divide all entries by to get

  5. Example Inverse of a 2x2 • Find the inverse of = (-2)(7) - (4)(-3) = -2 ≠ 0 2. Change signs: 1. Interchange: 3. Divide:

  6. Matrix With No Inverse • A matrix has no inverse if = ad - bc = 0.

  7. Example Inverse or No Inverse • Use to determine which matrix has an inverse. has no inverse. has an inverse.

  8. Solving a Matrix Equation • Solving a Matrix EquationIf the matrix A has an inverse, then the solution of the matrix equation • AX = B is given by X = A-1B.

  9. Example Solving a Matrix Equation • Use a matrix equation to solve The matrix form of the equation is

  10. Summary Section 2.4 - Part 1 • The inverse of a square matrix A is a square matrix A-1 with property that A-1A = I and AA-1 = I, where I is the identity matrix. • A 2x2 matrix has an inverse if • = ad - bc≠ 0. If so, the inverse matrix is

  11. Summary Section 2.4 - Part 2 • A system of linear equations can be written in the form AX = B, where A is a rectangular matrix of coefficients of the variables, X is a column of variables, and B is a column matrix of the constants from the right side of the system. If the matrix A has an inverse, then the solution of the equation is given by X = A-1B.

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