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MATH1120 Lecture Notes - Lecture 4: Invertible Matrix, Identity Matrix, Main Diagonal

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25 May 2020

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University of Newcastle

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Recall that a square matrix is a matrix with the same number of rows as columns. 0 where a is any square matrix of order n. the identity matrix, i, of order n is the n n matrix with 1"s on the main diagonal and all other entries 0, i. e. i. The identity matrix has the property that for any square matrix of order n, a, I is the only matrix that satisfies this property. Given the square matrix a, if there exists a square matrix b such that. Ab ba i then we call the matrix b the inverse of a and write. If matrix b is the inverse of matrix a then matrix a is the inverse of matrix b, i. e. 1a. Given a square matrix a to find its inverse we need to find a matrix. Let"s begin by considering the 2 2 case. 1a such where a, b, c and d are given.

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Related Questions

This is Linear Algebra, please help!

Please help with all questions. Will rate ASAP.

This question addresses inverse transformations from a conceptual standpoint, a. Suppose that a linear transformation T: x rightarrow Ax from A to A takes some nonzero vector x to 0. Can this transformation be reversed? Does A exist? Suppose that a linear transformation T: x rightarrow Ax from is not one-to-one. What does that tell you about the set of vectors b that can be produced by applying the transformation T to all vectors x in A? How do you know, in that case, that the transformation T is not reversible (i.e. that there does not exist an inverse transformation with the standard matrix A-1 that can be applied to the result of Ax to restore the vector x to its original value)? Suppose that a linear transformation T: x rightarrow Ax from A to A is not onto. What does that tell you about the set of vectors b that can be produced by applying the transformation T to all vectors x in A? Is it possible for a linear transformation x rightarrow Ax from A to A to be one-to-one but not onto, or onto but not one-to-one? If so, please give examples of each case. If not, explain why not. Is it possible for an arbitrary linear transformation x rightarrow Ax from A to B to be one-to-one but not onto, or onto but not one-to-one? If so. please give examples of each case. If not, explain why not. For what values of {a. b. c, d. e.f. a. h. i.j } will the matrix A = invertible? what is the least amount of partitioning that needs to be applied to M; to make M1 and M2 conformable for block multiplication? What is the product M = M1, M2 that you obtain using block multiplication with these partitions? What is the product M = M1 M2. that you obtain by multiplying the two matrices without partitioning? For which three values of a is the matrix B = not invertible, and why If M1 and M2 arc partitioned matrices such that M1 For each matrix product shown below, draw in the partitions that would need to be added to make the multiplication possible: Describe in general how the sizes of each of the partitions(#rows, #columns) in a matrix product C = AB are defined by the sizes of the partitions in each of the matrices A and B when partitioned matrices are multiplied.