Find A^2 (the product A A ) and A^3 (the prod.u c t(A^2) A). A=[ 2 -1 1 0 1

step 1 Given this matrix A, we want to find A squared and a cubed. So to find A square, we would have t

Find A^2 (the product A A ) and A^3 (the prod.u c t(A^2) A)....

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Key Concepts

Matrix Multiplication

Matrix multiplication is the process of combining two matrices by taking the dot product of the rows of the first matrix with the columns of the second matrix. This operation is fundamental in linear algebra and is used to represent composition of linear transformations, solve systems of linear equations, and compute matrix products in various applications.

Matrix Exponentiation

Matrix exponentiation involves raising a square matrix to a power, which means multiplying the matrix by itself repeatedly. This concept is useful in solving difference equations, analyzing linear dynamical systems, and computing functions of matrices. The process relies on successive applications of matrix multiplication and uses properties like associativity for simplification.

Transcript

00:01 Given this matrix a, we want to find a squared and a cubed.

00:05 So to find a square, we would have to times a by a.

00:09 So let's do that now as i write out the matrix.

00:13 Okay.

00:13 So when we're multiplying matrices to the same dimensions, you'll end up with the same dimension matrix.

00:18 So we're about 3 by 3 by 3 by 3.

00:20 So we're going to write up with the 3 by 3.

00:21 And if i wanted to find the first, see here, first row, first column, you take the first row from the first matrix and the first column for the second one.

00:30 So you're going to have two lots of two and then you've got multiplied by zeros after, so you've just got a four.

00:35 Equally, if we wanted to do now the second row first column, now you need one to second row, first column, and it's just going to be zero, zero to two, one lot zero, zero to zero.

00:44 And finally, third row, first column is going to be a big fat zero as you've got zero lots of those twos, three lots, zero lots of those threes and zero lots of those twos.

00:53 Okay now, so we're going to be moving on to our second column now, first row.

00:57 Two lots of that minus one is going to give us minus, minus two, minus one lot of that minus one is minus one, plus one lot of that three.

01:04 So we've got minus three plus three.

01:07 That's clearly going to be a zero...

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