Problem 25 - Algebra and Trigonometry

Question

Let $$ A=\left[\begin{array}{rr} {-3} & {-7} \\ {2} & {-9} \\ {5} & {0} \end{array}\right] \text { and } B=\left[\begin{array}{rr} {-5} & {-1} \\ {0} & {0} \\ {3} & {-4} \end{array}\right] $$ Solve each matrix equation for X. $$ 4 A+3 B=-2 X $$

Step-by-Step Solution

Answer

The matrix X is \(\left[\begin{array}{rr}{13.5} & {15.5} \ {-4} & {18} \ {-14.5} & {6}\end{array}\right]\)

Step 1: Multiply matrices A and B by their coefficients

Multiplying A by 4 and B by 3 we get: \(4A =\left[\begin{array}{rr}{-12} & {-28} \ {8} & {-36} \ {20} & {0}\end{array}\right]\) and \(3B = \left[\begin{array}{rr}{-15} & {-3} \ {0} & {0} \ {9} & {-12}\end{array}\right]\)

Step 2: Add the resulting matrices

Adding these two matrices gives us a new matrix: \[4A + 3B = \left[\begin{array}{rr}{-27} & {-31} \ {8} & {-36} \ {29} & {-12}\end{array}\right]\]

Step 3: Divide by -2 to find X

Since \(4A + 3B = -2X\), we divide both sides by -2, which gives us the matrix X: \[X = −\frac{1}{2} * (4A + 3B) = \left[\begin{array}{rr}{13.5} & {15.5} \ {-4} & {18} \ {-14.5} & {6}\end{array}\right]\]

Key Concepts

Matrix Multiplication

Matrix multiplication is a fundamental operation where two matrices are multiplied together to create a new matrix. The number of columns in the first matrix must equal the number of rows in the second matrix for the multiplication to be possible.

• Element Calculation: The element at the intersection of row "i" of the first matrix and column "j" of the second matrix is calculated as the sum of the products of elements from row "i" and column "j" corresponding to each element.
• Size of Resultant Matrix: If matrix A is of size "m x n" and matrix B is of size "n x p", the resultant matrix will have dimensions "m x p".

For example, when multiplying the matrix A by 4, each element in A is multiplied by 4, resulting in a new matrix of the same dimensions.

Matrix Addition

Matrix addition involves adding corresponding elements of two matrices to form a new matrix. This can only be performed on matrices of the same size.

• Element-wise Addition: Add each element from the first matrix to the corresponding element in the second matrix.
• Same Dimensions Required: Both matrices, A and B, must have the same number of rows and columns.

In the step-by-step exercise, we see matrix addition when adding the resultant matrices from the scalar multiplication of A and B, resulting in a matrix expressed as \(4A + 3B\).

Scalar Multiplication

Scalar multiplication is a straightforward process where each entry of a matrix is multiplied by a constant, known as the scalar.

• Individual Element Multiplication: Multiply each element of the matrix by the scalar.
• Retains Matrix Dimension: The resulting matrix will have the same dimensions as the original matrix.

In the given problem, both matrices A and B were subjected to scalar multiplication with 4 and 3 respectively, which modifies each element by this factor.

Matrix Division

Matrix division is not as straightforward as scalar division. Instead of direct division, it involves using an inverse or a division of scalars.

• Using Inverses: One common technique is multiplying by the inverse of a matrix. However, not all matrices have inverses.
• Division by Scalars: More common is the division of matrices by a scalar by multiplying the matrix by the reciprocal of the scalar.

In the provided exercise, division by a scalar was performed to solve for matrix X. The operation involved dividing the result of matrix addition by -2, which is equivalent to multiplying by -\(\frac{1}{2}\). This calculates the matrix X efficiently.