Determine The Value Of $m_{33}$ If You Multiplied Matrix $M$ By A Scalar Of -2.$\[ M = \begin{bmatrix} 3 & -5 & 8 & 12 \\ 0 & 11 & 7 & 3 \\ -4 & 9 & -2 & 0 \end{bmatrix} \\]A. -6 B. -4 C. 4 D. 0
Introduction
In linear algebra, matrices are used to represent systems of equations and perform various operations such as multiplication, addition, and scalar multiplication. When a matrix is multiplied by a scalar, each element of the matrix is multiplied by that scalar. In this article, we will determine the value of $m_{33}$ if the matrix $M$ is multiplied by a scalar of -2.
Understanding Matrix Multiplication
Matrix multiplication is a fundamental operation in linear algebra that involves multiplying two matrices to produce another matrix. The resulting matrix has the same number of rows as the first matrix and the same number of columns as the second matrix. When multiplying two matrices, each element of the resulting matrix is calculated by multiplying the corresponding elements of the rows of the first matrix and the columns of the second matrix.
Scalar Multiplication
Scalar multiplication is a special type of matrix multiplication where a matrix is multiplied by a scalar. In this operation, each element of the matrix is multiplied by the scalar. The resulting matrix has the same dimensions as the original matrix, but each element is scaled by the scalar value.
Determining the Value of $m_{33}$
To determine the value of $m_{33}$ after multiplying the matrix $M$ by a scalar of -2, we need to perform scalar multiplication on the matrix. The matrix $M$ is given by:
{ M = \begin{bmatrix} 3 & -5 & 8 & 12 \\ 0 & 11 & 7 & 3 \\ -4 & 9 & -2 & 0 \end{bmatrix} \}
To multiply this matrix by a scalar of -2, we multiply each element of the matrix by -2.
{ -2M = \begin{bmatrix} -6 & 10 & -16 & -24 \\ 0 & -22 & -14 & -6 \\ 8 & -18 & 4 & 0 \end{bmatrix} \}
From this resulting matrix, we can see that the value of $m_{33}$ is 4.
Conclusion
In conclusion, when the matrix $M$ is multiplied by a scalar of -2, the value of $m_{33}$ is 4. This is because each element of the matrix is multiplied by the scalar, resulting in a new matrix with the same dimensions but scaled elements.
Answer
The correct answer is C. 4.
Related Topics
- Matrix multiplication
- Scalar multiplication
- Linear algebra
Example Use Cases
- Determining the value of a matrix element after scalar multiplication
- Performing matrix operations such as addition and subtraction
- Solving systems of linear equations using matrices
Further Reading
- Linear Algebra by Gilbert Strang
- Matrix Algebra by James E. Gentle
- Introduction to Linear Algebra by David C. Lay
Q: What is matrix multiplication?
A: Matrix multiplication is a fundamental operation in linear algebra that involves multiplying two matrices to produce another matrix. The resulting matrix has the same number of rows as the first matrix and the same number of columns as the second matrix.
Q: What is scalar multiplication?
A: Scalar multiplication is a special type of matrix multiplication where a matrix is multiplied by a scalar. In this operation, each element of the matrix is multiplied by the scalar.
Q: How do I perform matrix multiplication?
A: To perform matrix multiplication, you need to multiply the corresponding elements of the rows of the first matrix and the columns of the second matrix. The resulting matrix has the same number of rows as the first matrix and the same number of columns as the second matrix.
Q: How do I perform scalar multiplication?
A: To perform scalar multiplication, you need to multiply each element of the matrix by the scalar.
Q: What is the difference between matrix multiplication and scalar multiplication?
A: The main difference between matrix multiplication and scalar multiplication is that matrix multiplication involves multiplying two matrices, while scalar multiplication involves multiplying a matrix by a scalar.
Q: Can I multiply a matrix by a vector?
A: Yes, you can multiply a matrix by a vector. This operation is called matrix-vector multiplication.
Q: What is the result of multiplying a matrix by a vector?
A: The result of multiplying a matrix by a vector is a vector.
Q: Can I multiply a vector by a matrix?
A: Yes, you can multiply a vector by a matrix. This operation is called vector-matrix multiplication.
Q: What is the result of multiplying a vector by a matrix?
A: The result of multiplying a vector by a matrix is a vector.
Q: What is the difference between matrix multiplication and vector-matrix multiplication?
A: The main difference between matrix multiplication and vector-matrix multiplication is that matrix multiplication involves multiplying two matrices, while vector-matrix multiplication involves multiplying a vector by a matrix.
Q: Can I multiply two vectors?
A: No, you cannot multiply two vectors. However, you can multiply a vector by a scalar.
Q: What is the result of multiplying a vector by a scalar?
A: The result of multiplying a vector by a scalar is a vector.
Q: Can I multiply a matrix by a matrix?
A: Yes, you can multiply a matrix by a matrix. This operation is called matrix-matrix multiplication.
Q: What is the result of multiplying a matrix by a matrix?
A: The result of multiplying a matrix by a matrix is a matrix.
Q: What is the difference between matrix multiplication and matrix-matrix multiplication?
A: The main difference between matrix multiplication and matrix-matrix multiplication is that matrix multiplication involves multiplying a matrix by a vector, while matrix-matrix multiplication involves multiplying a matrix by another matrix.
Q: Can I multiply a matrix by a matrix of different dimensions?
A: No, you cannot multiply a matrix by a matrix of different dimensions. The number of columns in the first matrix must be equal to the number of rows in the second matrix.
Q: What is the result of multiplying a matrix by a matrix of different dimensions?
A: The result of multiplying a matrix by a matrix of different dimensions is undefined.
Q: Can I multiply a matrix by a scalar of zero?
A: Yes, you can multiply a matrix by a scalar of zero. The result is a matrix of zeros.
Q: What is the result of multiplying a matrix by a scalar of zero?
A: The result of multiplying a matrix by a scalar of zero is a matrix of zeros.
Q: Can I multiply a matrix by a scalar of one?
A: Yes, you can multiply a matrix by a scalar of one. The result is the original matrix.
Q: What is the result of multiplying a matrix by a scalar of one?
A: The result of multiplying a matrix by a scalar of one is the original matrix.
Q: Can I multiply a matrix by a negative scalar?
A: Yes, you can multiply a matrix by a negative scalar. The result is a matrix with the opposite sign of the original matrix.
Q: What is the result of multiplying a matrix by a negative scalar?
A: The result of multiplying a matrix by a negative scalar is a matrix with the opposite sign of the original matrix.
Q: Can I multiply a matrix by a complex scalar?
A: Yes, you can multiply a matrix by a complex scalar. The result is a matrix with complex elements.
Q: What is the result of multiplying a matrix by a complex scalar?
A: The result of multiplying a matrix by a complex scalar is a matrix with complex elements.