Functions Involving Parameters, Vectors, and Matrices

Product AB is invertible.

Product AB results in zero-matrix.

A and B are identical matrices.

Make sure it's not invertible.

$(a + b)(c + d)$

$ab + cd$

$ac - bd$

$ad - bc$

Assumes inverses do not exist and implies communality relates only to the original pair.

Incorrectly assumes the order matters when it comes to existence and incorrectly suggests that nature impacts the ability to compute inverses.

Assuming existence affects the relationship between the pairs and incorrectly suggests that nature impacts the ability to compute inverses.

Their inverses also commute.

$a_{ik} + b_{kj}$

$e_{ij} = (a_{i} * b_{j}) + (b_{j} * a_{i})$

$a_{ij} * b_{ij} = c_{ij}$

$c_{ij} = \sum_{k=1}^{c} (a_{ik} * b_{kj})$

Multiplying a row by -1.

Multiplying a row by zero.

Swapping two rows.

Adding one row to another.

Matrix addition with a different-sized matrix

Matrix multiplication with a nonconformable matrix

Scalar multiplication

Augmenting matrices together

Rotation by 90 degrees counterclockwise.

Scaling by a factor of two.

Reflection across the x-axis.

Rotation by 180 degrees counterclockwise.

The magnitude decreases but direction remains same

No change occurs to vectors multiplied

The magnitude increases but direction remains same

The orientation is reversed (reflected through origin)

A 3x3 matrix.

A 3x2 matrix.

A 2x2 matrix.

A 2x3 matrix.

Matrix $B$.

The identity matrix.

The zero matrix.

Matrix $A$.