Statement 1 Let A=[ 8 -4 1 10 0 6 8 1 6 ]. Then A^-1=A^T

Statement 1 Let A=[ 8 -4 1 10 0 6...

Key Concepts

Invertible Matrix

An invertible matrix is a square matrix that has an inverse, meaning there exists another matrix that when multiplied with the original yields the identity matrix. This occurs if and only if the determinant of the matrix is nonzero, making it possible to solve systems of linear equations uniquely.

Matrix Transpose

The transpose of a matrix is formed by swapping its rows with its columns, so the element in the i-th row and j-th column becomes the element in the j-th row and i-th column. This operation is essential in many areas of linear algebra, affecting properties related to symmetry, orthogonality, and more.

Property Relating Inverse and Transpose

A fundamental property in linear algebra is that for any invertible matrix, the inverse of its transpose is equal to the transpose of its inverse. This relationship is formalized as (A^T)?¹ = (A?¹)^T and is key in manipulating and proving results about matrices.

Orthogonal Matrix

An orthogonal matrix is a type of square matrix for which the transpose is equal to its inverse. This equivalence implies that the matrix preserves lengths and angles, making it useful in representing rotations and reflections in space.

Transcript

00:01 So in the given question we have two statements that are given in the question and in the first statement we have a matrix a which is given as 8 minus 4 1, 10, 06 and 816.

00:23 So this is what we have.

00:27 And we are told that if a is this matrix, then the inverse of a is equal to its transpose.

00:37 And in the second statement what we are told is that if the matrix a is invertible, if matrix a is invertible, then the transpose of a is also in a.

00:55 Invertible.

00:57 This also invertible.

01:01 So this is what we have as the statement 1 and 2.

01:06 Right.

01:07 So when we look at these statements first what we should do is if the matrix a is invertible then it should follow that the determinant of a should be not equal to so if a matrix has a determinant equal to 0 then it won't be invertible so for it to be invertible the determinant should be not equal to 0 and if the determinant of a is not equal to 0 we can say that the the matrix the transpose matrix is also invertible right so if a is a matrix that is invertible then a transpose is also invertible since the determinant of a transpose is the same as the determinant of a, which would both be not equal to 0.

02:05 So this means a transpose, if a is invertible, a transpose is also invertible, which means we can say that the statement 2 is a true statement, right? so statement 2 is true.

02:20 Next up, let's check the first statement.

02:25 Statement and in the first statement we have a matrix right so let's take matrix a which is 8 minus 4 1 1006 and 816 and now let's take the transpose of this matrix which would be equal to 8 minus 4 1 1006 816 so we just we just wrote the columns, the rows of a as the columns of the new matrix which is the transpose of a...

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