Functions Involving Parameters, Vectors, and Matrices

Finding the inverse of a matrix.

Multiplication by a scalar matrix.

Matrix subtraction.

Matrix addition.

Scalar multiplication

Cross product

Dot product

Matrix addition

It is simply doubled.

It becomes zero unless it was initially zero.

It is multiplied by $2^n$, where $n$ is the size of the square matrix.

It remains unchanged.

The determinant indicates that negative values are present within the real-world data.

Adjusting any value within the model will always result in increased accuracy.

A single unique solution exists for all possible outcomes within the scenario.

The scenario modeled may have no unique solution or an infinite number of solutions.

$\frac{\pi}{2}$ radians

$\pi$ radians

$4\pi$ radians

$2\pi$ radians

Recording temperature changes hourly during a day at various weather stations.

Tracking inventory counts across multiple locations when new shipments have yet to affect stock levels.

Calculating depreciation on equipment assets over time across several departments.

Adjusting recipe ingredients based on changing serving sizes in multiple restaurants.

This modification introduces uniform skewing along both axes during transformations.

The modified matrix no longer maintains vector lengths or orientations during transformations.

All vectors get scaled uniformly by 'k' after being transformed by this modified matrix.

Only vectors aligned with either axis are affected by this change when transformed.

Add A to its transpose and then find the eigenvalues.

Multiply matrix A by its determinant and then apply Cramer’s Rule.

Subtract the identity matrix from A and solve for the null space.

Use the inverse of matrix A to find the solution for the variables.

They cancel each other out to produce a zero matrix when multiplied.

They result in a diagonal matrix with non-zero entries on the diagonal when multiplied.

They yield the identity matrix when multiplied in either order, $BB^{-1}$ or $B^{-1}B$.

Multiplying them results in another square matrix with identical elements as $B$.

Negative elements represent data entry errors as growth rates cannot be negative.

Negative elements in $G_0$ must be set to zero before calculating $P_n$ because they don't make sense contextually.

Positive elements after raising $G_0^n$ mean some populations eventually became positive again.

Negative elements indicate that there was a percentage decrease in population for those specific entries.