If { B = \begin{bmatrix} 8 & 3 \ -4 & -4 \end{bmatrix} $}$ And { C = \begin{bmatrix} 7 & 3 \ 6 & -5 \end{bmatrix} $}$, What Is { BC $}$?If The Matrix Exists, Select Its Size Before Entering Your Answer. If The Matrix

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Introduction

In linear algebra, matrix multiplication is a fundamental operation that allows us to combine two matrices to form a new matrix. Given two matrices A and B, the product AB is a new matrix whose elements are calculated by multiplying the rows of A by the columns of B. In this article, we will explore how to find the product of two matrices, specifically the product of matrices B and C.

Matrix B and Matrix C

Let's start by defining the two matrices B and C:

Matrix B

B = \begin{bmatrix} 8 & 3 \\ -4 & -4 \end{bmatrix}

Matrix C

C = \begin{bmatrix} 7 & 3 \\ 6 & -5 \end{bmatrix}

Matrix Multiplication

To find the product BC, we need to multiply the rows of matrix B by the columns of matrix C. The resulting matrix will have the same number of rows as matrix B and the same number of columns as matrix C.

Calculating the Product BC

To calculate the product BC, we need to perform the following calculations:

  • Multiply the first row of matrix B by the first column of matrix C:
    • (8)(7) + (3)(6) = 56 + 18 = 74
    • (8)(3) + (3)(-5) = 24 - 15 = 9
  • Multiply the first row of matrix B by the second column of matrix C:
    • (8)(3) + (3)(-5) = 24 - 15 = 9
    • (8)(-5) + (3)(-5) = -40 - 15 = -55
  • Multiply the second row of matrix B by the first column of matrix C:
    • (-4)(7) + (-4)(6) = -28 - 24 = -52
    • (-4)(3) + (-4)(-5) = -12 + 20 = 8
  • Multiply the second row of matrix B by the second column of matrix C:
    • (-4)(3) + (-4)(-5) = -12 + 20 = 8
    • (-4)(-5) + (-4)(-5) = 20 + 20 = 40

The Product Matrix BC

The resulting product matrix BC is:

BC = \begin{bmatrix} 74 & 9 \ -52 & 8 \end{bmatrix}

Matrix Size

The product matrix BC has the same number of rows as matrix B (2 rows) and the same number of columns as matrix C (2 columns). Therefore, the size of the product matrix BC is 2x2.

Conclusion

In this article, we have explored how to find the product of two matrices, specifically the product of matrices B and C. We have calculated the product matrix BC by multiplying the rows of matrix B by the columns of matrix C. The resulting product matrix BC has a size of 2x2 and has the following elements:

BC = \begin{bmatrix} 74 & 9 \ -52 & 8 \end{bmatrix}

References

  • Linear Algebra and Its Applications, 4th Edition, by Gilbert Strang
  • Matrix Algebra, 2nd Edition, by James E. Gentle

Further Reading

  • Matrix Multiplication: A Tutorial
  • Linear Algebra: A Comprehensive Introduction
  • Matrix Algebra: A Brief Introduction
    Matrix Multiplication: A Q&A Guide =====================================

Introduction

In our previous article, we explored how to find the product of two matrices, specifically the product of matrices B and C. In this article, we will answer some frequently asked questions about matrix multiplication.

Q: What is matrix multiplication?

A: Matrix multiplication is a fundamental operation in linear algebra that allows us to combine two matrices to form a new matrix. The resulting matrix is calculated by multiplying the rows of the first matrix by the columns of the second matrix.

Q: What are the conditions for matrix multiplication to be possible?

A: For matrix multiplication to be possible, the number of columns in the first matrix must be equal to the number of rows in the second matrix. In other words, if we have two matrices A and B, the product AB is possible only if the number of columns in A is equal to the number of rows in B.

Q: How do I calculate the product of two matrices?

A: To calculate the product of two matrices, you need to multiply the rows of the first matrix by the columns of the second matrix. The resulting matrix will have the same number of rows as the first matrix and the same number of columns as the second matrix.

Q: What is the size of the product matrix?

A: The size of the product matrix is determined by the number of rows in the first matrix and the number of columns in the second matrix. In other words, if we have two matrices A and B, the product AB will have the same number of rows as A and the same number of columns as B.

Q: Can I multiply a matrix by a scalar?

A: Yes, you can multiply a matrix by a scalar. When you multiply a matrix by a scalar, each element of the matrix is multiplied by the scalar.

Q: What is the difference between matrix multiplication and scalar multiplication?

A: Matrix multiplication is the operation of combining two matrices to form a new matrix, while scalar multiplication is the operation of multiplying each element of a matrix by a scalar.

Q: Can I multiply two matrices that have different sizes?

A: No, you cannot multiply two matrices that have different sizes. The number of columns in the first matrix must be equal to the number of rows in the second matrix for matrix multiplication to be possible.

Q: What is the identity matrix in matrix multiplication?

A: The identity matrix is a special matrix that has the property that when it is multiplied by any matrix, the resulting matrix is the same as the original matrix. The identity matrix is denoted by I and has 1s on the main diagonal and 0s elsewhere.

Q: Can I multiply a matrix by its transpose?

A: Yes, you can multiply a matrix by its transpose. When you multiply a matrix by its transpose, the resulting matrix is called the dot product or inner product of the matrix.

Q: What is the transpose of a matrix?

A: The transpose of a matrix is a new matrix that is obtained by interchanging the rows and columns of the original matrix.

Conclusion

In this article, we have answered some frequently asked questions about matrix multiplication. We have discussed the conditions for matrix multiplication to be possible, how to calculate the product of two matrices, and the size of the product matrix. We have also discussed scalar multiplication, the identity matrix, and the transpose of a matrix.

References

  • Linear Algebra and Its Applications, 4th Edition, by Gilbert Strang
  • Matrix Algebra, 2nd Edition, by James E. Gentle

Further Reading

  • Matrix Multiplication: A Tutorial
  • Linear Algebra: A Comprehensive Introduction
  • Matrix Algebra: A Brief Introduction