Let A be any 3 ×2 matrix. Then AA^T is (a) null matrix (b) a diagonal matrix (c) an identity mat

Let A be any 3 ×2 matrix. Then AA^T is (a) null matrix (b) a...

Key Concepts

Matrix Multiplication

Matrix multiplication is the process of multiplying two matrices by taking the dot product of rows from the first matrix and columns from the second matrix. This operation is defined when the number of columns in the first matrix equals the number of rows in the second, resulting in a product matrix with dimensions corresponding to the outer dimensions of the multiplied matrices.

Matrix Transpose

The transpose of a matrix is formed by switching its rows with its columns. For any given matrix, this operation flips the matrix over its diagonal, and it has important properties when combined with matrix multiplication, especially in the creation of symmetric matrices.

Symmetric Matrix

A symmetric matrix is a square matrix that is equal to its own transpose; that is, the element in the ith row and jth column is identical to the element in the jth row and ith column for all indices. When a matrix is multiplied by its transpose, as in the product AA^T, the resulting matrix is always symmetric.

Transcript

00:01 So in the given question we are told that we have a matrix a, we have a matrix a which is of the order 3 by 2, right? and we have to find what kind of matrix we can get with a times a transpose.

00:24 So what we have as options is a times a transpose would be a matrix it would be a diagonal matrix a diagonal matrix it is an identity matrix or option d which says it is a symmetric matrix so we should find which of these options is correct, right? so what we can do over here is we know the order of a, right? so since we know the order of a, let's take a matrix a in which we have three rows and two columns, right? so let's write that assume a is equal to a b, c, d, f, that is a matrix with three rows and two columns.

01:30 Then a transpose would be equal to a -c -e -b -d -f, right? so now let's take the product of a -and -a -transpose.

01:42 So a times a transpose would then be equal to when we take the product of these two matrices, what we would get is a square plus b squared, ac plus bd, ae plus bf, ac plus bd, c square plus d square, c square, ce plus d square, ce plus df, ae plus df, ae plus bf, ce plus df, ce plus df, c e plus d.

02:20 And e square plus x squared so this is what we get as the product of a times a transpose right so from this we can say that the matrix a times a transpose is not a matrix right it is not equal to zero and we can also say that it is not a diagonal matrix also right so we can say these and we can also say that a times a transpose is not the identity matrix and this is what we are left with, right? so now what we can do is we can take the transpose of this matrix and check whether it is symmetric or not, right? so when we take the transpose of a times a transpose, we again get the same matrix which is a square plus b square.

03:22 Ac plus bd, ae plus ae plus bf, ac plus bd, c squared plus d square, ce plus df, and the third column is ae plus bf, ce plus df, e squared plus f squared...

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