Select The Correct Answer.${ A=\left[\begin{array}{rr} 1 & -4 \ -3 & 2 \ 7 & -5 \end{array}\right] \quad \text{and} \quad B=\left[\begin{array}{rr} 1 & 7 \ -2 & 11 \end{array}\right] }$Which Of The Following Is The Product Matrix
Introduction
Matrix multiplication is a fundamental concept in linear algebra, and it plays a crucial role in various fields such as physics, engineering, and computer science. In this article, we will explore the concept of matrix multiplication, its properties, and how to perform it. We will also provide a step-by-step guide on how to multiply two matrices and discuss the different types of matrix multiplication.
What is Matrix Multiplication?
Matrix multiplication is a mathematical operation that takes two matrices as input and produces another matrix as output. The resulting matrix is obtained by multiplying the elements of the rows of the first matrix with the elements of 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.
Properties of Matrix Multiplication
Matrix multiplication has several properties that make it a powerful tool in linear algebra. Some of the key properties of matrix multiplication include:
- Associativity: Matrix multiplication is associative, meaning that the order in which we multiply the matrices does not affect the result.
- Distributivity: Matrix multiplication is distributive, meaning that we can multiply a matrix by a scalar and then multiply the result by another matrix.
- Commutativity: Matrix multiplication is not commutative, meaning that the order in which we multiply the matrices affects the result.
How to Multiply Two Matrices
To multiply two matrices, we need to follow these steps:
- Check if the matrices can be multiplied: Before multiplying two matrices, we need to check if they can be multiplied. This means that the number of columns in the first matrix must be equal to the number of rows in the second matrix.
- Identify the dimensions of the matrices: Once we have checked that the matrices can be multiplied, we need to identify the dimensions of the matrices. The resulting matrix will have the same number of rows as the first matrix and the same number of columns as the second matrix.
- Multiply the elements of the rows and columns: To multiply the elements of the rows and columns, we need to multiply the elements of the rows of the first matrix with the elements of the columns of the second matrix.
- Calculate the resulting matrix: Once we have multiplied the elements of the rows and columns, we need to calculate the resulting matrix.
Step-by-Step Guide to Multiplying Two Matrices
Let's consider an example to illustrate the steps involved in multiplying two matrices. Suppose we have two matrices A and B, where:
A=
B=
To multiply these matrices, we need to follow the steps outlined above.
- Check if the matrices can be multiplied: The number of columns in matrix A is 2, and the number of rows in matrix B is 2. Therefore, the matrices can be multiplied.
- Identify the dimensions of the matrices: The resulting matrix will have 3 rows (the same number of rows as matrix A) and 2 columns (the same number of columns as matrix B).
- Multiply the elements of the rows and columns: To multiply the elements of the rows and columns, we need to multiply the elements of the rows of matrix A with the elements of the columns of matrix B.
- Calculate the resulting matrix: Once we have multiplied the elements of the rows and columns, we need to calculate the resulting matrix.
The resulting matrix is:
C=
C=
C=
Conclusion
Matrix multiplication is a fundamental concept in linear algebra, and it plays a crucial role in various fields such as physics, engineering, and computer science. In this article, we have explored the concept of matrix multiplication, its properties, and how to perform it. We have also provided a step-by-step guide on how to multiply two matrices and discussed the different types of matrix multiplication.
Frequently Asked Questions
- What is matrix multiplication? Matrix multiplication is a mathematical operation that takes two matrices as input and produces another matrix as output.
- What are the properties of matrix multiplication? Matrix multiplication has several properties, including associativity, distributivity, and commutativity.
- How do I multiply two matrices? To multiply two matrices, you need to follow these steps: check if the matrices can be multiplied, identify the dimensions of the matrices, multiply the elements of the rows and columns, and calculate the resulting matrix.
References
- Linear Algebra and Its Applications by Gilbert Strang
- Matrix Algebra by James E. Gentle
- Introduction to Linear Algebra by Gilbert Strang
Matrix Multiplication Q&A ==========================
Frequently Asked Questions
Q: What is matrix multiplication?
A: Matrix multiplication is a mathematical operation that takes two matrices as input and produces another matrix as output. The resulting matrix is obtained by multiplying the elements of the rows of the first matrix with the elements of the columns of the second matrix.
Q: What are the properties of matrix multiplication?
A: Matrix multiplication has several properties, including:
- Associativity: Matrix multiplication is associative, meaning that the order in which we multiply the matrices does not affect the result.
- Distributivity: Matrix multiplication is distributive, meaning that we can multiply a matrix by a scalar and then multiply the result by another matrix.
- Commutativity: Matrix multiplication is not commutative, meaning that the order in which we multiply the matrices affects the result.
Q: How do I multiply two matrices?
A: To multiply two matrices, you need to follow these steps:
- Check if the matrices can be multiplied: Before multiplying two matrices, you need to check if they can be multiplied. This means that the number of columns in the first matrix must be equal to the number of rows in the second matrix.
- Identify the dimensions of the matrices: Once you have checked that the matrices can be multiplied, you need to identify the dimensions of the matrices. The resulting matrix will have the same number of rows as the first matrix and the same number of columns as the second matrix.
- Multiply the elements of the rows and columns: To multiply the elements of the rows and columns, you need to multiply the elements of the rows of the first matrix with the elements of the columns of the second matrix.
- Calculate the resulting matrix: Once you have multiplied the elements of the rows and columns, you need to calculate the resulting matrix.
Q: What is the difference between matrix multiplication and scalar multiplication?
A: Matrix multiplication and scalar multiplication are two different operations. Scalar multiplication involves multiplying a matrix by a scalar (a number), while matrix multiplication involves multiplying two matrices together.
Q: Can I multiply a matrix by a vector?
A: Yes, you can multiply a matrix by a vector. This is known as matrix-vector multiplication. The resulting vector will have the same number of elements as the number of columns in the matrix.
Q: Can I multiply a vector by a matrix?
A: Yes, you can multiply a vector by a matrix. This is known as vector-matrix multiplication. The resulting vector will have the same number of elements as the number of rows in the matrix.
Q: What is the transpose of a matrix?
A: The transpose of a matrix is a new matrix that is obtained by swapping the rows and columns of the original matrix. The transpose of a matrix is denoted by the symbol T.
Q: How do I find the transpose of a matrix?
A: To find the transpose of a matrix, you need to swap the rows and columns of the original matrix. This can be done using a variety of methods, including using a computer program or by hand.
Q: What is the inverse of a matrix?
A: The inverse of a matrix is a new matrix that is obtained by reversing the order of the rows and columns of the original matrix. The inverse of a matrix is denoted by the symbol A^(-1).
Q: How do I find the inverse of a matrix?
A: To find the inverse of a matrix, you need to use a variety of methods, including using a computer program or by hand. One common method is to use the Gauss-Jordan elimination algorithm.
Q: What is the determinant of a matrix?
A: The determinant of a matrix is a scalar value that is obtained by multiplying the elements of the matrix together. The determinant of a matrix is denoted by the symbol det(A).
Q: How do I find the determinant of a matrix?
A: To find the determinant of a matrix, you need to use a variety of methods, including using a computer program or by hand. One common method is to use the Laplace expansion formula.
Q: What is the rank of a matrix?
A: The rank of a matrix is the maximum number of linearly independent rows or columns in the matrix.
Q: How do I find the rank of a matrix?
A: To find the rank of a matrix, you need to use a variety of methods, including using a computer program or by hand. One common method is to use the Gauss-Jordan elimination algorithm.
Q: What is the null space of a matrix?
A: The null space of a matrix is the set of all vectors that are mapped to the zero vector by the matrix.
Q: How do I find the null space of a matrix?
A: To find the null space of a matrix, you need to use a variety of methods, including using a computer program or by hand. One common method is to use the Gauss-Jordan elimination algorithm.
Q: What is the column space of a matrix?
A: The column space of a matrix is the set of all linear combinations of the columns of the matrix.
Q: How do I find the column space of a matrix?
A: To find the column space of a matrix, you need to use a variety of methods, including using a computer program or by hand. One common method is to use the Gauss-Jordan elimination algorithm.
Q: What is the row space of a matrix?
A: The row space of a matrix is the set of all linear combinations of the rows of the matrix.
Q: How do I find the row space of a matrix?
A: To find the row space of a matrix, you need to use a variety of methods, including using a computer program or by hand. One common method is to use the Gauss-Jordan elimination algorithm.
Q: What is the orthogonal complement of a matrix?
A: The orthogonal complement of a matrix is the set of all vectors that are orthogonal to the columns of the matrix.
Q: How do I find the orthogonal complement of a matrix?
A: To find the orthogonal complement of a matrix, you need to use a variety of methods, including using a computer program or by hand. One common method is to use the Gram-Schmidt process.
Q: What is the singular value decomposition (SVD) of a matrix?
A: The singular value decomposition (SVD) of a matrix is a factorization of the matrix into three matrices: U, Σ, and V.
Q: How do I find the SVD of a matrix?
A: To find the SVD of a matrix, you need to use a variety of methods, including using a computer program or by hand. One common method is to use the power iteration method.
Q: What is the eigenvalue decomposition (EVD) of a matrix?
A: The eigenvalue decomposition (EVD) of a matrix is a factorization of the matrix into two matrices: D and V.
Q: How do I find the EVD of a matrix?
A: To find the EVD of a matrix, you need to use a variety of methods, including using a computer program or by hand. One common method is to use the power iteration method.
Q: What is the QR decomposition of a matrix?
A: The QR decomposition of a matrix is a factorization of the matrix into two matrices: Q and R.
Q: How do I find the QR decomposition of a matrix?
A: To find the QR decomposition of a matrix, you need to use a variety of methods, including using a computer program or by hand. One common method is to use the Gram-Schmidt process.
Q: What is the LU decomposition of a matrix?
A: The LU decomposition of a matrix is a factorization of the matrix into two matrices: L and U.
Q: How do I find the LU decomposition of a matrix?
A: To find the LU decomposition of a matrix, you need to use a variety of methods, including using a computer program or by hand. One common method is to use the Doolittle method.
Q: What is the Cholesky decomposition of a matrix?
A: The Cholesky decomposition of a matrix is a factorization of the matrix into two matrices: L and L^T.
Q: How do I find the Cholesky decomposition of a matrix?
A: To find the Cholesky decomposition of a matrix, you need to use a variety of methods, including using a computer program or by hand. One common method is to use the Cholesky algorithm.
Q: What is the LDL decomposition of a matrix?
A: The LDL decomposition of a matrix is a factorization of the matrix into three matrices: L, D, and L^T.
Q: How do I find the LDL decomposition of a matrix?
A: To find the LDL decomposition of a matrix, you need to use a variety of methods, including using a computer program or by hand. One common method is to use the LDL algorithm.
Q: What is the LDLT decomposition of a matrix?
A: The LDLT decomposition of a matrix is a factorization of the matrix into three matrices: L, D, and L^T.
Q: How do I find the LDLT decomposition of a matrix?
A: To find the LDLT decomposition of a matrix, you need to use a variety of methods, including using a computer program or by hand. One common method is to use the LDLT algorithm.
Q: What is the Bunch-Kaufman decomposition of a matrix?
A: The Bunch-Kaufman decomposition of a matrix is a factorization of the matrix into two matrices: L and U.
Q: How do I find the Bunch-Kaufman decomposition of a matrix?
A: