Question

Use the following matrices. Determine whether the given expression is defined. If it is defined, express the result as a single matrix; if it is not, write "not defined" $$ A=\left[\begin{array}{rrr} 0 & 3 & -5 \\ 1 & 2 & 6 \end{array}\right] \quad B=\left[\begin{array}{rrr} 4 & 1 & 0 \\ -2 & 3 & -2 \end{array}\right] \quad C=\left[\begin{array}{rr} 4 & 1 \\ 6 & 2 \\ -2 & 3 \end{array}\right] $$ -3 B

Short answer

The result is \( \begin{bmatrix} -12 & -3 & 0 \ 6 & -9 & 6 o \right] \)

Step-by-step solution

Understand the Expression

The given expression is o negative three times matrix B (i.e., -3B). We need to determine if we can perform this operation and what the result will be if it is defined.

Verify Matrix Compatibility

Matrix B is a 2x3 (2 rows and 3 columns) matrix. To multiply a scalar (in this case, -3) with a matrix, ensure the dimensions of the matrix remain unchanged. Matrix B has dimensions that allow scalar multiplication.

Scalar Multiplication of Matrix B

Multiply each element of matrix B by -3. Matrix B is: o B = o \( \begin{bmatrix} 4 & 1 & 0 \ -2 & 3 & -2 o \right] \). Let's perform the multiplication:o -3 * 4 = -12o -3 * 1 = -3o -3 * 0 = 0o -3 * -2 = 6o -3 * 3 = -9o -3 * -2 = 6o The resulting matrix is:\( \begin{bmatrix} -12 & -3 & 0 \ 6 & -9 & 6 o \right] \)

Key concepts

Scalar Multiplication

Scalar multiplication in matrix operations involves multiplying every element of a matrix by a scalar value, which is simply a single number. The key thing to remember for scalar multiplication is that it only changes the values within the matrix, not its dimensions. For example, given matrix B:
\[ B = \begin{bmatrix} 4 & 1 & 0 \ -2 & 3 & -2 \end{bmatrix} \] and the scalar \(-3\), we multiply each element of B by \(-3\):
\[ -3 \cdot 4 = -12 \] \[ -3 \cdot 1 = -3 \] \[ -3 \cdot 0 = 0 \] \[ -3 \cdot (-2) = 6 \] \[ -3 \cdot 3 = -9 \] \[ -3 \cdot (-2) = 6 \]
After performing these operations, we get the resulting matrix:
\[ -3B = \begin{bmatrix} -12 & -3 & 0 \ 6 & -9 & 6 \end{bmatrix} \]
This result shows that the dimensions of the matrix remain the same while each element is scaled by \(-3\).

Matrix Dimensions

Understanding matrix dimensions is essential for performing various matrix operations. Matrix dimensions are specified as \(m \times n\), where \(m\) is the number of rows and \(n\) is the number of columns. For instance, matrix B is a 2x3 matrix because it has 2 rows and 3 columns:
\[ B = \begin{bmatrix} 4 & 1 & 0 \ -2 & 3 & -2 \end{bmatrix} \]
Matrix dimensions are critical in determining the feasibility of operations like addition, subtraction, and multiplication. For scalar multiplication, as seen in the provided exercise, the dimensions of the matrix do not change. The resultant matrix from scalar multiplication will always have the same dimensions as the original matrix. This consistency is highly beneficial because it simplifies the process, ensuring that only the values within the matrix are altered.

Defined Expressions

In matrix operations, ensuring that the expression is defined is crucial before performing any calculation. A defined expression means that the operation can be legally and correctly carried out according to matrix algebra rules. In the given exercise, the expression \(-3B\) is defined because scalar multiplication is always permitted regardless of the matrix's dimensions.
For more complicated operations, such as matrix addition or multiplication of two matrices, checking if expressions are defined involves ensuring:

• The matrices have appropriate dimensions for addition/subtraction (i.e., they must be of the same dimensions).
  
• For matrix multiplication, the number of columns in the first matrix must equal the number of rows in the second matrix.
  

Once these conditions are verified, the expression can be successfully evaluated. If not, the operation is 'not defined,' meaning no valid result can be computed.