How to Find the Inverse of a 3x3 Matrix: Step-by-Step Guide
Learn how to find the inverse of a 3x3 matrix with clear steps and key definitions. Understanding the determinant, matrix of minors, cofactors, and completing the process to find the inverse matrix efficiently.
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Teachings for Teachings for Exercise 6E Exercise 6E
? ? ? ? ? ? ? ? ?? ? ?+ ?? ? = ?? ? ? ? ? Matrices ? ? ? ?= ?? ?? You need to be able to find the inverse of a 3x3 Matrix This consists of a number of steps. Although your calculator can do this, if the Matrix contains unknowns then you will need to follow the steps You also need to know some key definitions which we will see as we go through the steps 6E
? ? ? ? ? ? ? ? ?? ? ?+ ?? ? = ?? ? ? ? ? Matrices ? ? ? ?= ?? ?? You need to be able to find the inverse of a 3x3 Matrix If the 3x3 Matrix we are finding the inverse of is ?, then: Find the determinant of ? Form the matrix of minors of ? (use ? to represent this) 3) From the matrix of minors, form the matrix of cofactors (?) according to the pattern below + + + 1) 2) + + 4) Write down the transpose of the matrix of cofactors, ?? 5) The inverse of A is then given by: 1 ? 1= ?????? 6E
? ? ? ? ? ? ? ? ?? ? ?+ ?? ? = ?? ? ? ? ? Matrices ? ? ? ?= ?? ?? 1 0 2 3 4 1 1 0 You need to be able to find the inverse of a 3x3 Matrix , find ? 1 Given that the Matrix ? = 1 If the 3x3 Matrix we are finding the inverse of is ?, then: ???? = 1 ? ? ? ? ? ? ? ? ?? ? ?+ ?? ? = ?? ? ? ? ? Find the determinant of ? Form the matrix of minors of ? (use ? to represent this) 3) From the matrix of minors, form the matrix of cofactors (?) according to the pattern below + + + 1) Replace values 2) 4 1 0 30 1 0+ 10 4 Calculate each determinant = 1 1 2 2 1 = 1 0 + 1 3 0 2 + 1(0 8) Calculate + + = 1 4) Write down the transpose of the matrix of cofactors, ?? 5) The inverse of A is then given by: 1 ? 1= ?????? 6E
? ? ? ? ? ? ? ? ?? ? ?+ ?? ? = ?? ? ? ? ? Matrices ? ? ? ?= ?? ?? 1 0 2 3 4 1 1 0 You need to be able to find the inverse of a 3x3 Matrix , find ? 1 Given that the Matrix ? = 1 If the 3x3 Matrix we are finding the inverse of is ?, then: ???? = 1 4 1 0 1 0 0 2 1 2 1 0 1 0 1 0 1 1 0 2 1 2 4 1 3 1 3 4 1 3 1 3 4 Find the determinant of ? Form the matrix of minors of ? (use ? to represent this) 3) From the matrix of minors, form the matrix of cofactors (?) according to the pattern below + + + 1) 2) 1 0 1 1 Calculate determinants 1 1 2 2 1 8 7 4 + + ? = 1 Form the matrix of cofactors 4) Write down the transpose of the matrix of cofactors, ?? 5) The inverse of A is then given by: 1 2 8 7 4 ? = 1 1 2 1 1 ? 1= ?????? 6E
? ? ? ? ? ? ? ? ?? ? ?+ ?? ? = ?? ? ? ? ? Matrices ? ? ? ?= ?? ?? 1 0 2 3 4 1 1 0 You need to be able to find the inverse of a 3x3 Matrix , find ? 1 Given that the Matrix ? = 1 If the 3x3 Matrix we are finding the inverse of is ?, then: ???? = 1 1 2 8 7 4 Find the determinant of ? Form the matrix of minors of ? (use ? to represent this) 3) From the matrix of minors, form the matrix of cofactors (?) according to the pattern below + + + 1 1 4 8 7 4 1) ? = 1 1 2 1 Write down the transpose by swapping rows and columns 2) 1 2 1 2 7 1 1 4 ??= 8 1 1 2 2 8 7 1 1 1 + + ? = ??= 1 2 2 The transpose of a Matrix is created by interchanging the rows and columns 1 4 2 3 The transpose of Matrix ? is written as ?? 4) Write down the transpose of the matrix of cofactors, ?? 5) The inverse of A is then given by: 1 4 2 3 1 ? 1= ?????? 6E
? ? ? ? ? ? ? ? ?? ? ?+ ?? ? = ?? ? ? ? ? Matrices ? ? ? ?= ?? ?? 1 0 2 3 4 1 1 0 You need to be able to find the inverse of a 3x3 Matrix , find ? 1 Given that the Matrix ? = 1 If the 3x3 Matrix we are finding the inverse of is ?, then: ???? = 1 1 ? 1= ?????? Find the determinant of ? Form the matrix of minors of ? (use ? to represent this) 3) From the matrix of minors, form the matrix of cofactors (?) according to the pattern below + + + 8 7 4 1) Replace both 1 2 1 2 7 1 1 4 2) 1 ? 1= 1 8 Multiply by the fraction value outside 1 2 8 1 2 1 1 ? 1= 1 2 1 2 1 1 + + ??= 7 4 4) Write down the transpose of the matrix of cofactors, ?? 5) The inverse of A is then given by: Remember that although your calculator can do this, you need to know the steps if you get an algebraic version (see example 18 in the textbook!) 1 ? 1= ?????? 6E
? ? ? ? ? ? ? ? ?? 1= ? 1? 1 ?? ? ?+ ?? ? = ?? ? ? ? ? Matrices ? ? ? ?= ?? ?? You need to be able to find the inverse of a 3x3 Matrix ? = ?? 1 Multiply both sides by ?? (at the start) (??)? = (??) ?? 1 The right hand side is the identity matrix The matrices ? and ? are non-singular. Prove that ?? 1= ? 1? 1. ??? = ? Multiply both sides by ? 1 at the start There will be some terms which we can eliminate ? 1??? = ? 1? Start by letting ? = ?? 1 ?? = ? 1 Multiply both sides by ? 1 at the start ? 1?? = ? 1? 1 We can simplify ? = ? 1? 1 Replace C with the initial assumption ?? 1= ? 1? 1 This is a useful result that you should learn! 6E
? ? ? ? ? ? ? ? ?? 1= ? 1? 1 ?? ? ?+ ?? ? = ?? ? ? ? ? Matrices ? ? ? ?= ?? ?? You need to be able to find the inverse of a 3x3 Matrix ? 1= ? Multiply both sides by A at the start ?? 1= ?? 2 0 1 3 1 3 0 2 Rewrite both sides The matrix ? = and the matrix ? = ?2 1 8 17 10 5 9 ? is such that ?? 1= So we can prove the initial statement by showing that ?2= ? 5 3 6 4 2 0 1 3 1 3 0 2 2 0 1 3 1 3 0 2 a) Show that ? 1= ? b) Find ? 1 1 1 Calculate each part 4 + 0 3 0 + 0 + 0 2 + 0 + 2 6 + 3 + 3 0 + 1 + 0 3 1 2 6 + 0 6 0 + 0 + 0 3 + 0 + 4 = Simplify 1 0 0 0 1 0 0 0 1 = 6E
? ? ? ? ? ? ? ? ?? 1= ? 1? 1 ?? ? ?+ ?? ? = ?? ? ? ? ? Matrices ? ? ? ?= ?? ?? 8 17 10 5 9 You need to be able to find the inverse of a 3x3 Matrix ?? 1= 5 3 6 4 You can use the result above to rewrite the left side 2 0 1 3 1 3 0 2 The matrix ? = and the matrix 8 17 10 5 9 ? 1? 1= 5 3 6 4 1 8 17 10 5 9 Multiply by ? at the ends ? is such that ?? 1= 5 3 6 4 8 17 10 5 9 ? 1? 1? = ? 5 3 6 4 We can simplify the left side a) Show that ? 1= ? b) Find ? 1 8 17 10 5 9 ? 1= ? 5 3 6 4 Write in Matrix A 8 17 10 5 9 2 0 1 3 1 3 0 2 ? 1= 5 3 6 4 1 You can do the multiplication 7 4 2 2 1 0 6 3 1 ? 1= 6E
