Question

Matrices \(A\) and \(B\) are given below. Simplify the given expression. $$ A=\left[\begin{array}{cc} 1 & -1 \\ 7 & 4 \end{array}\right] \quad B=\left[\begin{array}{cc} -3 & 2 \\ 5 & 9 \end{array}\right] $$ $$ 3 A-A $$

Short answer

\( \begin{bmatrix} 2 & -2 \\ 14 & 8 \end{bmatrix} \)

Step-by-step solution

Calculate 3A

To find \(3A\), multiply each entry in matrix \(A\) by 3:\[3 \times \begin{bmatrix} 1 & -1 \ 7 & 4 \end{bmatrix} = \begin{bmatrix} 3 \times 1 & 3 \times -1 \ 3 \times 7 & 3 \times 4 \end{bmatrix} = \begin{bmatrix} 3 & -3 \ 21 & 12 \end{bmatrix}\] This is the matrix \(3A\).

Calculate A

Matrix \(A\) is given as:\[A = \begin{bmatrix} 1 & -1 \ 7 & 4 \end{bmatrix}\] This is matrix \(A\) that we use to perform subtraction in the next step.

Subtract A from 3A

Now subtract matrix \(A\) from \(3A\):\[3A - A = \begin{bmatrix} 3 & -3 \ 21 & 12 \end{bmatrix} - \begin{bmatrix} 1 & -1 \ 7 & 4 \end{bmatrix} = \begin{bmatrix} 3-1 & -3-(-1) \ 21-7 & 12-4 \end{bmatrix}\]Simplifying each element, we get:\[\begin{bmatrix} 2 & -2 \ 14 & 8 \end{bmatrix}\] This is the final simplified expression for \(3A - A\).

Key concepts

Matrix Multiplication

Matrix multiplication involves the multiplication of a matrix by another matrix or a scalar (a single number). When multiplying a matrix by a scalar, each element of the matrix is multiplied by that scalar.
For example, to multiply matrix \( A \) by 3, you perform the following steps for each element in \( A \):

• Multiply the element in the first row, first column \( 1 \) by 3 to get \( 3 \).
• Multiply the element in the first row, second column \( -1 \) by 3 to get \( -3 \).
• Multiply the element in the second row, first column \( 7 \) by 3 to get \( 21 \).
• Finally, multiply the element in the second row, second column \( 4 \) by 3 to get \( 12 \).

The resulting matrix, \( 3A \), becomes: \[\begin{bmatrix}3 & -3 \21 & 12\end{bmatrix}\] The key here is consistency: every element in the original matrix has been scaled by the same factor.

Matrix Subtraction

Matrix subtraction is the process of subtracting corresponding elements of two matrices of the same size. It is similar to arithmetic subtraction but is done element-wise.
You subtract the elements in one matrix from the elements in the corresponding positions of another matrix.
For example, when you subtract matrix \( A \) from \( 3A \), the operations are as follows:

• From the first row, first column, subtract \( 1 \) from \( 3 \) to get \( 2 \).
• For the first row, second column, subtract \( -1 \) from \( -3 \) to get \( -2 \).
• In the second row, first column, subtract \( 7 \) from \( 21 \) to get \( 14 \).
• Finally, in the second row, second column, subtract \( 4 \) from \( 12 \) to get \( 8 \).

This operation results in a new matrix: \[\begin{bmatrix}2 & -2 \14 & 8\end{bmatrix}\] Matrix subtraction is straightforward, just ensure that the matrices are the same size.

Matrix Simplification

Matrix simplification refers to reducing a matrix expression to its simplest form. This involves performing any necessary arithmetic operations to combine and simplify terms.
In the context of solving expressions like \( 3A - A \), simplification involves using matrix multiplication and subtraction rules to achieve the final simplified matrix.
The process may include:

• First, perform scalar multiplication if needed, as was the case in creating \( 3A \).
• Next, carry out subtraction by subtracting matrix \( A \) from \( 3A \), as described above.
• Check all operations to ensure accuracy for each element in the matrices.
• Finally, verify that the resulting matrix is in its simplest possible form and corresponds correctly to the operations performed.

In this context, the simplified result of \( 3A - A \) gives a neat matrix solution: \[\begin{bmatrix}2 & -2 \14 & 8\end{bmatrix}\]