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QABE_Notes06

# QABE_Notes06 - QABE Lecture 6 Matrices II The Inverse...

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QABE Lecture 6 Matrices II: The Inverse & Determinant in Small Matrices School of Economics, UNSW 2011 Contents 1 Introduction 1 2 The Inverse 2 2.1 Defined . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 3 Determinant Excursus 3 3.1 ‘Small’ Determinants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 3.2 Determinants of higher orders . . . . . . . . . . . . . . . . . . . . . . . . . 3 3.3 Cofactors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 4 The Inverse Really 5 4.1 The Adjoint of a Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 4.2 The Inverse by the Adjoint . . . . . . . . . . . . . . . . . . . . . . . . . . 5 4.3 A Useful Check! . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 5 On Linear Equations 7 5.1 The problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1 Introduction This lecture is a significant step up in diﬃculty for our matrix algebra, but it is also a significant improvement in the power of our matrix toolbox. We begin by coming up with an equivalent operation to division in arithmetic, known as the matrix inverse . It does just the same job (a matrix multiplied by its inverse equals the identity matrix (the stand-in for ‘1’ in matrix algebra)), but like other matrix operations, has its own specific rules and properties. Following from this, we are caused to wonder how we might actually come up with the inverse? Afterall, it isn’t much use unless we can find out what its value is. Whilst there are a few techniques for obtaining the inverse (e.g. row reduction) we will consider just one, the adjoint method . Now this is itself a little bit tricky, even on small matrices, but it will introduce us to a very important property of any square matrix, known as the determinant , and the adjoint method will give us an idea about why sometimes we can get an inverse, and sometimes we can’t. More on this in the next lecture. Although all of what we do in this lecture is applicable to much larger matrices (and is where these techniques have real power), we won’t be doing them on anything 1

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ECON 1202/ECON 2291: QABE c School of Economics, UNSW bigger than a two-by-two or three-by-three. Next lecture we’ll unleash the power of the computer on our matrix world, and so do some tackle some really big problems.
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QABE_Notes06 - QABE Lecture 6 Matrices II The Inverse...

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