Question Find the size of matrix $B$ if $A$ is a $3 \times 8$ matrix and $A(B B)$ is defined.

Studdy Solution

STEP 1

1. Matrix multiplication is defined such that if $A$ is an $m \times n$ matrix and $B$ is a $p \times q$ matrix, then $A$ and $B$ can be multiplied (i.e., $AB$ is defined) if and only if $n = p$ .
2. The product of an $m \times n$ matrix and a $p \times q$ matrix is an $m \times q$ matrix.
3. The notation $A(BB)$ suggests that matrix $B$ is being multiplied by itself and then the result is being multiplied by matrix $A$ .

STEP 2

1. Determine the necessary condition for the multiplication $BB$ to be defined.
2. Determine the size of matrix $B$ based on the multiplication $A(BB)$ being defined.
3. Conclude the dimensions of matrix $B$ .

STEP 3

Consider the necessary condition for $BB$ to be defined. For a matrix $B$ to be multiplied by itself, it must be square, meaning it has the same number of rows and columns.
$B \text{ is an } n \times n \text{ matrix}$

STEP 4

Since $A$ is a $3 \times 8$ matrix and $A(BB)$ is defined, the product $BB$ must result in a matrix with 8 rows for $A$ to be able to multiply with it.
$BB \text{ is an } 8 \times n \text{ matrix}$

STEP 5

From the condition that $BB$ is an $8 \times n$ matrix and $B$ is square, we deduce that $B$ must be an $8 \times 8$ matrix.

STEP 6

Conclude that the dimensions of matrix $B$ are $8 \times 8$ .
Matrix $B$ is $8 \times 8$ .

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Question Find the size of matrix $B$ if $A$ is a $3 \times 8$ matrix and $A(B B)$ is defined.

Studdy Solution

STEP 1

STEP 2

1. Determine the necessary condition for the multiplication $BB$ to be defined.
2. Determine the size of matrix $B$ based on the multiplication $A(BB)$ being defined.
3. Conclude the dimensions of matrix $B$ .

STEP 3

Consider the necessary condition for $BB$ to be defined. For a matrix $B$ to be multiplied by itself, it must be square, meaning it has the same number of rows and columns.
$B \text{ is an } n \times n \text{ matrix}$

STEP 4

Since $A$ is a $3 \times 8$ matrix and $A(BB)$ is defined, the product $BB$ must result in a matrix with 8 rows for $A$ to be able to multiply with it.
$BB \text{ is an } 8 \times n \text{ matrix}$

STEP 5

From the condition that $BB$ is an $8 \times n$ matrix and $B$ is square, we deduce that $B$ must be an $8 \times 8$ matrix.

STEP 6

Conclude that the dimensions of matrix $B$ are $8 \times 8$ .
Matrix $B$ is $8 \times 8$ .

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Question Find the size of matrix BBB if AAA is a 3×83 \times 83×8 matrix and A(BB)A(B B)A(BB) is defined.

Studdy Solution

STEP 1

STEP 2

1. Determine the necessary condition for the multiplication BBBBBB to be defined.2. Determine the size of matrix BBB based on the multiplication A(BB)A(BB)A(BB) being defined.3. Conclude the dimensions of matrix BBB.

STEP 3

Consider the necessary condition for BBBBBB to be defined. For a matrix BBB to be multiplied by itself, it must be square, meaning it has the same number of rows and columns.B is an n×n matrix B \text{ is an } n \times n \text{ matrix} B is an n×n matrix

STEP 4

Since AAA is a 3×83 \times 83×8 matrix and A(BB)A(BB)A(BB) is defined, the product BBBBBB must result in a matrix with 8 rows for AAA to be able to multiply with it.BB is an 8×n matrix BB \text{ is an } 8 \times n \text{ matrix} BB is an 8×n matrix

STEP 5

From the condition that BBBBBB is an 8×n8 \times n8×n matrix and BBB is square, we deduce that BBB must be an 8×88 \times 88×8 matrix.

STEP 6

Conclude that the dimensions of matrix BBB are 8×88 \times 88×8.Matrix BBB is 8×88 \times 88×8.

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Question Find the size of matrix $B$ if $A$ is a $3 \times 8$ matrix and $A(B B)$ is defined.

1. Determine the necessary condition for the multiplication $BB$ to be defined.
2. Determine the size of matrix $B$ based on the multiplication $A(BB)$ being defined.
3. Conclude the dimensions of matrix $B$ .

Consider the necessary condition for $BB$ to be defined. For a matrix $B$ to be multiplied by itself, it must be square, meaning it has the same number of rows and columns.
$B \text{ is an } n \times n \text{ matrix}$

Since $A$ is a $3 \times 8$ matrix and $A(BB)$ is defined, the product $BB$ must result in a matrix with 8 rows for $A$ to be able to multiply with it.
$BB \text{ is an } 8 \times n \text{ matrix}$

From the condition that $BB$ is an $8 \times n$ matrix and $B$ is square, we deduce that $B$ must be an $8 \times 8$ matrix.

Conclude that the dimensions of matrix $B$ are $8 \times 8$ .
Matrix $B$ is $8 \times 8$ .