Write down the inverse of each of the following elementary matrices: (a) ( 0 1 1

step 1 Hello, Shwantz, we need to find the elementary matrix. We need to use our elementary matrix to f

Write down the inverse of each of the following elementary...

Key Concepts

Permutation Matrix

A permutation matrix is a type of elementary matrix that results from swapping two rows of an identity matrix. Such matrices are particularly simple because they are their own inverses; applying the same row swap operation twice restores the original order of rows.

Inverse of an Elementary Matrix

The inverse of an elementary matrix is also an elementary matrix corresponding to the inverse row operation. This means that for each elementary row operation, there is a reverse operation that can undo it, such as switching rows back, multiplying by the reciprocal of a scaling factor, or subtracting a previously added multiple of one row from another.

Row Addition (Shear) Matrix

A row addition matrix is obtained by adding a multiple of one row to another row in the identity matrix. Its inverse is formed by performing the opposite operation, which involves subtracting the same multiple of the original row, thus ensuring that the combined effect of the operation and its inverse is the identity transformation.

Elementary Matrix

An elementary matrix is obtained by applying a single elementary row operation to an identity matrix. These matrices are fundamental in linear algebra because they represent the basic building blocks for performing row operations on matrices, and any matrix can be reduced to row echelon form by a sequence of such operations.

Elementary Row Operations

Elementary row operations include row swapping, row scaling (multiplying a row by a nonzero constant), and row addition (adding a multiple of one row to another). Each of these operations has a corresponding elementary matrix which, when multiplied with another matrix, effectively performs that row operation on it.

Transcript

00:01 Hello, shwantz, we need to find the elementary matrix.

00:04 We need to use our elementary matrix to find the inverse of this particular matrix, right? so first we need to calculate this matrix and then we are going to find our inverse.

00:15 So first we calculate first we multiply these two matrices, right? so this is three cross three matrix, this is three cross three matrix and this is three cross three matrix, right? so this row, this first row must be multiplied by this column.

00:28 This first row must be multiplied by this column so this is 1 and this is ab so i can write as 1 plus ab right now this is 1 and this is a b now if i look about look around second term would be this is 0 and this is this is 0 and this is a second 0 right and this if i talk about this term this term would be 0 and this is again 0 now if i talk about this term this term again zero and this is b right this can be written as b and this is zero and this is one and this is zero so this can be written as one so this is zero zero and zero again zero right and this is zero and this is zero and this is zero and again zero and zero and one right so zero zero and one must be multiplied by if i write this must be multiplied by one zero zero zero and zero c right now a is further calculated as if i write this this is one plus a b multiplied by this this is zero and zero so one plus ab and again if i write this this is one plus a b zero and this is again zero so this is again a and this is zero and this is zero and this is again zero right now this is b multiplied by one and one zero so this is zero and b multiplied by zero and this is one and this is zero right and this whole row multiplied by this column so zero zero and again zero right now in the next term i can write as zero and this is zero and this is zero and zero and zero and one into zero is zero is zero is zero is zero and one into zero is zero is and this is 0 against 0 and against c right it means we need to find the inverse of this matrix our a matrix is this so how can i find inverse i equals to a equals to ia right how can i find the inverse of this matrix by elementary method so i can write this matrix as a equals to ia it means if i need if i need to convert, if i need to find the inverse, then a must be converted to i so that a inverse multiplied by a we get...

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