Transpose of a Matrix

The video tutorial explains the concept of the transpose of a matrix, denoted as A dash or A raised to power T. It describes how to obtain the transpose by interchanging rows and columns or reflecting across the main diagonal. The tutorial covers properties of transpose, such as involution, scalar multiplication, and sum and product transposes. Examples are provided to verify these properties, including the transpose of a product being equal to the product of transposes in reverse order.

What is the result of transposing a matrix?

• The matrix is inverted.

• The matrix is multiplied by a scalar.

• Rows become columns and columns become rows.

• The matrix remains unchanged.

Which of the following is true about the transpose of a matrix?

• Transpose of a matrix is obtained by interchanging its rows and columns.

• Transpose of a matrix is the same as its inverse.

• Transpose of a matrix is obtained by multiplying it by a scalar.

• Transpose of a matrix is always a square matrix.

What is the property of the transpose of a matrix product?

• The transpose of a product is the product of the transposes in reverse order.

• The transpose of a product is the inverse of the product.

• The transpose of a product is the product of the transposes in the same order.

• The transpose of a product is the sum of the transposes.

How does the transpose of a scalar multiple of a matrix behave?

• It is equal to the scalar plus the transpose of the matrix.

• It is equal to the scalar times the transpose of the matrix.

• It is equal to the inverse of the matrix.

• It is equal to the transpose of the matrix divided by the scalar.

What is the result of taking the transpose of the transpose of a matrix?

• It results in the inverse of the matrix.

• It results in a scalar matrix.

• It results in a zero matrix.

• It results in the original matrix.

Which property is verified by showing that A plus B transpose equals A transpose plus B transpose?

• Involution property

• Distributive property

• Associative property

• Commutative property

What is the order of the matrix obtained by transposing a 3x1 column matrix?

• 3x1 column matrix

• 1x1 scalar matrix

• 3x3 square matrix

• 1x3 row matrix