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2x2 Matrices, Determinants and Inverses. Evaluating Determinants of 2x2 Matrices Using Inverse Matrices to Solve Equations. Evaluating Determinants of 2x2 Matrices.
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2x2 Matrices, Determinants and Inverses Evaluating Determinants of 2x2 Matrices Using Inverse Matrices to Solve Equations
Evaluating Determinants of 2x2 Matrices • When you multiply two matrices together, in the order AB orBA, and the result is the identity matrix, then matrices A and B are inverses. Identity matrix for multiplication
Evaluating Determinants of 2x2 Matrices You only have to prove ONE of these. • To show two matrices are inverses… • AB = IORBA = I • AA-1 = IORA-1A = I Inverse of A Inverse of A
Evaluating Determinants of 2x2 Matrices • Example 1: • Show that B is the multiplicative inverse of A.
Evaluating Determinants of 2x2 Matrices • Example 1: • Show that B is the multiplicative inverse of A.
Evaluating Determinants of 2x2 Matrices • Example 1: • Show that B is the multiplicative inverse of A. AB = I. Therefore, B is the inverse of A and A is the inverse of B.
Evaluating Determinants of 2x2 Matrices • Example 1: • Show that B is the multiplicative inverse of A. AB = I. Therefore, B is the inverse of A and A is the inverse of B. Check by multiplying BA…answer should be the same
Evaluating Determinants of 2x2 Matrices • Example 1: • Show that B is the multiplicative inverse of A. AB = I. Therefore, B is the inverse of A and A is the inverse of B. Check by multiplying BA…answer should be the same
Evaluating Determinants of 2x2 Matrices • Example 2: • Show that the matrices are multiplicative inverses.
Evaluating Determinants of 2x2 Matrices • Example 2: • Show that the matrices are multiplicative inverses. BA = I. Therefore, B is the inverse of A and A is the inverse of B.
Evaluating Determinants of 2x2 Matrices • The determinant is used to tell us if an inverse exists. • If det ≠ 0, an inverse exists. • If det = 0, no inverse exists. A Matrix with a determinant of zero is called a SINGULAR matrix
Evaluating Determinants of 2x2 Matrices • To calculate a determinant…
Evaluating Determinants of 2x2 Matrices • To calculate a determinant… Multiply along the diagonal
Evaluating Determinants of 2x2 Matrices • To calculate a determinant… Take the product of the leading diagonal, and subtract the product of the non-leading diagonal Equation to find the determinant
Evaluating Determinants of 2x2 Matrices • Example 1: Evaluate the determinant.
Evaluating Determinants of 2x2 Matrices • Example 1: Evaluate the determinant.
Evaluating Determinants of 2x2 Matrices • Example 1: Evaluate the determinant.
Evaluating Determinants of 2x2 Matrices • Example 1: Evaluate the determinant. det = -23 Therefore, there is an inverse.
Evaluating Determinants of 2x2 Matrices • Example 2: Evaluate the determinant.
Evaluating Determinants of 2x2 Matrices • Example 2: Evaluate the determinant.
Evaluating Determinants of 2x2 Matrices • Example 2: Evaluate the determinant. det = 0 Therefore, there is no inverse.
Evaluating Determinants of 2x2 Matrices • How do you know if a matrix has an inverse ANDwhat that inverse is? • Given , the inverse of A is given by: Equation to find an inverse matrix This is called the adjoint matrix. It is formed by interchanging elements in the leading diagonal and negating elements in the non-leading diagonal
Evaluating Determinants of 2x2 Matrices • Example 1: • Determine whether the matrix has an inverse. If an inverse exists, find it.
Evaluating Determinants of 2x2 Matrices • Example 1: • Determine whether the matrix has an inverse. If an inverse exists, find it. Step 1: Find det M
Evaluating Determinants of 2x2 Matrices • Example 1: • Determine whether the matrix has an inverse. If an inverse exists, find it. Step 1: Find det M det M = -2, the inverse of M exists.
Evaluating Determinants of 2x2 Matrices • Example 1: • Determine whether the matrix has an inverse. If an inverse exists, find it. Step 2: Find the adjoint matrix. i.e
Evaluating Determinants of 2x2 Matrices • Example 1: • Determine whether the matrix has an inverse. If an inverse exists, find it. Step 2: Find the adjoint matrix. i.e Change signs
Evaluating Determinants of 2x2 Matrices • Example 1: • Determine whether the matrix has an inverse. If an inverse exists, find it. Step 2: Find the adjoint matrix. i.e Change signs Adjoint of M
Evaluating Determinants of 2x2 Matrices • Example 1: • Determine whether the matrix has an inverse. If an inverse exists, find it. Step 2: Find the adjoint matrix. i.e Change positions Adjoint of M
Evaluating Determinants of 2x2 Matrices • Example 1: • Determine whether the matrix has an inverse. If an inverse exists, find it. Step 2: Find the adjoint matrix. i.e Change positions Adjoint of M
Evaluating Determinants of 2x2 Matrices • Example 1: • Determine whether the matrix has an inverse. If an inverse exists, find it. Step 3: Use the equation to find the inverse.
Evaluating Determinants of 2x2 Matrices • Example 1: • Determine whether the matrix has an inverse. If an inverse exists, find it. Step 3: Use the equation to find the inverse.
Evaluating Determinants of 2x2 Matrices • Example 2: • Determine whether the matrix has an inverse. If an inverse exists, find it.
Evaluating Determinants of 2x2 Matrices • Example 2: • Determine whether the matrix has an inverse. If an inverse exists, find it.
Evaluating Determinants of 2x2 Matrices • Example 2: • Determine whether the matrix has an inverse. If an inverse exists, find it.