Inverse of a 2 x 2 matrix and determinants
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Determinants
Inverse of a 2 ´2 matrix The inverse of a matrix, A, is the matrix which, when multiplied by A gives the identity matrix I. [Equation goes here - download the original to see it.] Example [Equation goes here - download the original to see it.] then [Equation goes here - download the original to see it.] Proof of inverse formula [Equation goes here - download the original to see it.] [Equation goes here - download the original to see it.] then [Equation goes here - download the original to see it.] [Equation goes here - download the original to see it.] [Equation goes here - download the original to see it.] Only square matrices have inverses. The inverses of 3 ´ 3 matrices are found by the technique of Gaussian row reduction or by a formula, which is included in a later section. The number [Equation goes here - download the original to see it.] is called the determinant of the square 2 x 2 matrix A. It is denoted by[Equation goes here - download the original to see it.] [Equation goes here - download the original to see it.] A determinant is a number, it is not a matrix. [Equation goes here - download the original to see it.] Determinant of a 3 ´ 3 matrix The determinant of a 3 x 3 matrix [Equation goes here - download the original to see it.] [Diagram goes here - download the original to see it. Example (3) [Equation goes here - download the original to see it.] Answer [Equation goes here - download the original to see it.] The definition seems rather cumbersome, but in practice can be quite easy to recall. Thus, in this example [Diagram goes here - download the original to see it.] [Equation goes here - download the original to see it.] Now proceed to the next column. We are dealing with the contribution to the determinant from the "2" in the second column. But first… [Diagram goes here - download the original to see it.] [Equation goes here - download the original to see it.] [Diagram goes here - download the original to see it.] [Equation goes here - download the original to see it.] Now proceed to the last column - in this example the "2" in the third column. [Diagram goes here - download the original to see it.] [Equation goes here - download the original to see it.] [Diagram goes here - download the original to see it.] [Equation goes here - download the original to see it.] [Equation goes here - download the original to see it.] Example (4) [Equation goes here - download the original to see it.]
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Contents of
Inverse of a 2 x 2 matrix and determinants
1 Determinants
2 Inverse matrices and matrix multiplication
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